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CHAPTER 6
Additional Topics in Trigonometry
Section 6.1 Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . 466
Section 6.2 Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . 473
Section 6.3 Vectors in the Plane . . . . . . . . . . . . . . . . . . . . 482
Section 6.4 Vectors and Dot Products . . . . . . . . . . . . . . . . . 495
Section 6.5 Trigonometric Form of a Complex Number . . . . . . . . 503
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
© Houghton Mifflin Company. All rights reserved.

CHAPTER 6
Additional Topics in Trigonometry
Section 6.1
Law of Sines
■
■
■
If ABC is any oblique triangle with sides a, b, and c, then the Law of Sines says
a
sin A
b
sin B
c
sin C .
You should be able to use the Law of Sines to solve an oblique triangle for the remaining three parts, given:
(a) Two angles and any side (AAS or ASA)
(b) Two sides and an angle opposite one of them (SSA)
1. If A is acute and h b sin A:
(a) a < h, no triangle is possible.
(b) a h or a ≥ b, one triangle is possible.
(c) h < a < b, two triangles are possible.
2. If A is obtuse and h b sin A:
(a) a ≤ b, no triangle is possible.
(b) a > b, one triangle is possible.
The area of any triangle equals one-half the product of the lengths of two sides times the sine of their
included angle.
A 1 2ab sin C 1 2ac sin B 1 2bc sin A
Vocabulary Check
1. oblique 2.
3. (a) Two; any; AAS; ASA 4.
(b) Two; an opposite; SSA
1. Given:
b
c
A 25, B 60, a 12
C 180 25 60 95
a
12
sin B sin 60 24.59 in.
sin A sin 25
a
12
sin C sin 95 28.29 in.
sin A sin 25
b
sin B
1
2 bc sin A; 1 2 ab sin C; 1 ac sin B
2
2. Given:
b
c
A 35, B 55, a 18
C 180 35 55 90
a
18
sin B sin 55 25.71 mm
sin A sin 35
a
18
sin C sin 90 31.38 mm
sin A sin 35
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466

Section 6.1 Law of Sines 467
3. Given:
a
b
B 15, C 125, c 20
A 180 15 125 40
c
20
sin A sin 40 15.69 cm
sin C sin 125
c
20
sin B sin 15 6.32 cm
sin C sin 125
4. Given:
a
b
B 40, C 110, c 30
A 180 40 110 30
c
30
sin A sin 30 15.96 ft
sin C sin 110
c
30
sin B sin 40 20.52 ft
sin C sin 110
5. Given:
C 180 80 15 25 30 74 15
a
c
6. Given:
c
A 80 15, B 25 30, b 2.8
b
sin B sin A 2.8
sin 80 15 6.41 km
sin 25 30
b
sin B sin C 2.8
sin 74 15 6.26 km
sin 25 30
C 180 88 35 22 45 68 40
a
7. Given:
c
A 88 35, B 22 45, b 50.2
b
50.2
sin A sin 88 35 129.77 yd
sin B sin 22 45
b
50.2
sin C sin 68 40 120.92 yd
sin B sin 22 45
sin B b sin A
a
A 36, a 8, b 5
5 sin36
8
0.3674 ⇒ B 21.6
C 180 A B 180 36 21.6 122.4
a
8
sin C sin122.4 11.49
sin A sin36
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8. Given:
sin C c sin A
a
Case 1 Case 2
C 74.2
B 180 A C 45.8
b
9. Given:
b
c
A 60, a 9, c 10
10 sin 60
9
a 9 sin 45.8
sin B 7.45
sin A sin 60
A 102.4, C 16.7, a 21.6
0.9623 ⇒ C 74.2 or C 105.8
B 180 A C 180 102.4 16.7 60.9
a
21.6
sin B sin 60.9 19.32
sin A sin 102.4
a
21.6
sin C sin 16.7 6.36
sin A sin 102.4
C 105.8
B 180 A C 14.2
b
a 9 sin 14.2
sin B 2.55
sin A sin 60

468 Chapter 6 Additional Topics in Trigonometry
10. Given:
B 180 A C 101.1
a
b
A 24.3, C 54.6, c 2.68
c
2.68 sin 24.3
sin A 1.35
sin C sin 54.6
c
2.68 sin 101.1
sin B 3.23
sin C sin 54.6
11. Given:
sin B b sin A
a
C 180 A B 180 110 15 18 13 51 32
c
A 110 15, a 48, b 16
16 sin 110 15
48
0.31273 ⇒ B 18 13
a
sin A sin C 48
sin 51 32 40.06
sin 110 15
12. Given:
sin C c sin B
b
A 180 B C 174.68174 41
a
b
sin B
B 2 45, b 6.2, c 5.8
5.8 sin 2 45
0.04488 ⇒ C 2.572 34
6.2
6.2 sin 174 41
sin A 11.97
sin 2 45
13. Given:
sin B b sin A
a
C 180 A B 21.26
c
A 110, a 125, b 100
100 sin 110
125
0.75175 ⇒
a
125 sin 21.26
sin C 48.23
sin A sin 110
B 48.74
14. Given: A 110, a 125, b 200
A obtuse and a < b ⇒ No solution
16. Given:
sin B b sin A
a
c
A 76, a 34, b 21
21 sin 76
34
C 180 76 36.8 67.2
0.5993 ⇒ B 36.8
a
34
sin C sin 67.2 32.30
sin A sin 76
15. Given:
sin B b sin A
a
No solution
A 76, a 18, b 20
20 sin 76
18
1.078
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Section 6.1 Law of Sines 469
17. Given:
sin B b sin A
a
0.9522 ⇒ B 72.21 or 107.79
Case 1 Case 2
B 72.21
C 180 58 72.21 49.79
c
A 58, a 11.4, b 12.8
12.8 sin 58
11.4
C a
11.4
sin C sin 49.79 10.27
sin A sin 58
B 107.79
180 58 107.79 14.21
c
a
11.4
sin C sin 14.21 3.30
sin A sin 58
C
C
12.8
11.4
12.8
11.4
A
58° 72.21°
B
58° 107.79°
A
B
18. Given: A 58, a 4.5, b 12.8 19.
a < h b sin 58
4.5 < 10.86
No solution
Area 1 2 ab sin C
1 2 610 sin110
28.2 square units
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20. Area 1 2 ac sin B 21. Area 1 2bc sin A
22. A 5 15, b 4.5, c 22
1 29230 sin130
1 26785 sin38 45 Area 1 2bc sin A
1057.1 square units
1782.3 square units
1 24.522 sin 5.25
4.529 square units
23. Area 1 2 ac sin B 24. Area 1 2 ab sin C
1 210358 sin 75 15
1 21620 sin 85 45
2888.6 square units
159.6 square units
25. Angle CAB 70
Angle B 20 14 34
(a)
(b)
(c)
16
sin 70 h
sin 34
h
20°
h
14°
70°
16 sin 34
sin 70
34°
16
9.52 meters
26. (a)
(b)
20°50'
40 m
A 180 98 20.83 61.17 or 61 10
40
sin A h
sin 20 50
(c) h 16.2 m
98°
h
20 50 20.83
40 sin 20 50
⇒ h
sin 61.17

470 Chapter 6 Additional Topics in Trigonometry
27.
sin A a sin B
b
A 29.97
ACD 90 29.97 60
Bearing: S 60 W or ( 240 in plane navigation)
A
720
C
D
500 sin46
720
500
44°
46°
B
0.4995
28. Given:
a
A
46°
A 74 28 46,
B 180 41 74 65, c 100
C 180 46 65 69
c
100
sin A sin 46 77 meters
sin C sin 69
69°
C
100
65°
B
29. (a)
r
30. (a)
18.8°
17.5°
z
(b)
r
40°
r
3000 sin12180 40
sin 40
3000 ft
(c) s 40 4385.71 3061.80 feet
180
s
4385.71 feet
(b)
(c)
9000 ft y
Not drawn to scale
x
sin 17.5 9000
sin 1.3
x 119,289.1261 feet 22.6 miles
y
sin 71.2 x
sin 90
y x sin 71.2 119,289.1261 sin 71.2
112,924.963 feet 21.4 miles
(d) z 119,289.1261 sin 18.8 38,442.8 feet
31.
ACD 65
ADC 180 65 15 100
32.
CDB 180 100 80
B 180 80 70 30
a
c
b
30
sin A sin 15 15.53 km
sin B sin 30
b
30
sin C sin 135 42.43 km
sin B sin 30
A 20, 90 63 153, c 10
1
4 2.5
C 180 20 153 7
b
c
2.5 sin 153
sin B 9.31
sin C sin 7
65°
A
(Pine Knob)
A
d
80°
15°
30
2.5 B
b
(Colt Station)
C
a
65° 70°
D
a
B
(Fire)
C
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d b sin A 9.31 sin 20 3.2 miles

Section 6.1 Law of Sines 471
33.
sin42
10
sin42
42
sin 48
17
10 sin 48 0.4371
17
25.919
16.1
34. (a)
(b) Third angle in triangle
A
θ
2 mi
d
φ
B
Not drawn to scale
180 180 ⇒
d
2
sin sin
d 2 sin
sin
2 sin
sin
35. (a)
(b)
sin 5.45
58.36
5.36
d
58.36
sin sin
58.36
sin 1 d sin
0.0934 5.45 (c)
⇒ sin d sin
58.36
α
58.36
(d)
90 5.36 180 ⇒ 84.64
d sin 58.36
sin sin84.64 58.36
sin
d
10
20
30
40
50
60
324.1 154.2 95.2 63.8 43.3 28.1
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36. (a)
(c)
(e)
sin
sin
9 18
sin 0.5 sin
arcsin0.5 sin
arcsin0.5 sin
c
18
sin sin
c
c 18 sin
sin
18 sin arcsin0.5 sin
sin
(b)
(d)
1
0
−1
30
0
0
0.4 0.8 1.2 1.6 2.0 2.4 2.8
0.1960 0.3669 0.4848 0.5234 0.4720 0.3445 0.1683
25.95 23.07 19.19 15.33 12.29 10.31 9.27
37. False. If just the three angles are known, the
triangle cannot be solved.
Domain: 0 < <
Range: 0 < ≤ 6
Domain: 0 < <
Range: 9 < c < 27
→ 0, c →27.
As
As → , c → 9.
38. True. No angle could be 90.

472 Chapter 6 Additional Topics in Trigonometry
39. Yes, the Law of Sines can be used to solve a right
triangle if you are given at least one side and one
angle, or two sides.
40. Answers will vary. A 36, a 5
(a) b 4 one solution
(b) b 7 two solutions h b sin A < a < b
(c) b 10 no solution a < h b sin
41. Given:
b
c
A 45, B 52, a 16
C 180 45 52 83
a
16
sin B sin 52 17.83
sin A sin 45
a
16
sin C sin 83 22.46
sin A sin 45
a b sin
C
2 c cos A B
2
16 17.83 sin
83
2 22.46 cos 45 52
2
22.42 22.42
42. Given:
b
c
A 42, B 60, a 24
C 180 42 60 78
a
24
sin B sin 60 31.06
sin A sin 42
a
24
sin C sin 78 35.08
sin A sin 42
a b cos
C
2 c sin A B
2
24 31.06 cos
78
2 35.08 sin 42 60
2
5.49 5.49
43.
tan sin
cos
12 5
44.
cot 9 2
sec 13
5
sin 2
85 285 85
cot 5 12
csc 13
12
cos cot sin
9 2 2
85 9
85 985 85
sec 85
9
45. 6 sin 8 cos 3 6 1 2sin8 3 sin8 3 3sin 11 sin 5
46. 2 cos 2 cos 5 2 1 2cos2 5 cos2 5 cos 3 cos 7
47.
48.
5
3 cos sin
6
5
2
3 3 1 2 sin
6 5
3 2 sin 11
6 sin 3
2
3 2 1 2 1 9 4
5 4 cos
3 sin
5 4 cos
12 cos 19
12
12 cos 19
12
6 5
3
3 5
sin sin
4 6 5 2 1 2 cos 3
4 5
6 cos 3
4 5
6
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Section 6.2 Law of Cosines 473
Section 6.2
Law of Cosines
■
■
■
If ABC is any oblique triangle with sides a, b, and c, then the Law of Cosines says:
(a)
(b)
a 2 b 2 c 2 2bc cos A
b 2 a 2 c 2 2ac cos B
or
or
cos A b2 c 2 a 2
2bc
cos B a2 c 2 b 2
2ac
(c)
or cos C a2 b 2 c 2
c 2 a 2 b 2 2ab cos C
2ab
You should be able to use the Law of Cosines to solve an oblique triangle for the remaining three parts, given:
(a) Three sides (SSS)
(b) Two sides and their included angle (SAS)
Given any triangle with sides of lengths a, b, and c, then the area of the triangle is
Area ss as bs c, where s a b c . (Heron’s Formula)
2
Vocabulary Check
1. c 2 a 2 b 2 2ab cos C
2. Heron’s Area
3.
1
2bh, ss as bs c
1. Given:
sin B b sin A
a
a 12, b 16, c 18
cos A b2 c 2 a 2
2bc
162 18 2 12 2
21618
0.8712 ⇒ B 60.61
C 180 60.61 40.80 78.59
0.75694 ⇒ A 40.80
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2. Given:
sin C c sin A
a
a 8, b 18, c 12
cos A b2 c 2 a 2
2bc
182 12 2 8 2
21812
0.5312 ⇒ C 32.09
B 180 32.09 20.74 127.17
3. Given:
sin B b sin A
a
a 8.5, b 9.2, c 10.8
cos A b2 c 2 a 2
2bc
9.22 10.8 2 8.5 2
29.210.8
9.2 sin 49.51
8.5
C 180 55.40 49.51 75.09
0.9352 ⇒ A 20.74
0.6493 ⇒ A 49.51
0.82315 ⇒ B 55.40

474 Chapter 6 Additional Topics in Trigonometry
4. Given:
sin A a sin C
c
a 4.2, b 5.4, c 2.1
cos C a2 b 2 c 2
2ab
4.22 5.4 2 2.1 2
24.25.4
4.2 sin 20.8
2.1
B 180 20.85 45.38 113.77
0.7102 ⇒ A 45.38
0.9345 ⇒ C 20.85
5. Given:
a 10, c 15, B 20
b 2 a 2 c 2 2ac cos B 100 225 21015 cos 20 43.0922 ⇒ b 6.56 mm
cos A b2 c 2 a 2
2bc
43.0922 225 100
26.5615
C 180 20 31.40 128.60
0.8541 ⇒ A 31.40
6. Given:
a 16, c 10, B 40
b 2 a 2 c 2 2ac cos B 256 100 21610 cos 40 110.8658 ⇒ b 10.5293 10.53 km
cos C a2 b 2 c 2
2ab
A 180 40 37.62 102.38
256 110.8658 100
21610.53
0.7920 ⇒ C 37.62
7. Given:
a 10.4, c 12.5, B 50 30 50.5
b 2 a 2 c 2 2ac cos B 10.4 2 12.5 2 210.412.5 cos 50.5 99.0297 ⇒ b 9.95 ft
cos A b2 c 2 a 2
2bc
99.0297 12.52 10.4 2
29.9512.5
C 180 50.5 53.75 75.75 75 45
0.5914 ⇒ A 53.75 53 45
8. Given:
a 20.2, c 6.2, B 80 45 80.75
b 2 a 2 c 2 2ac cos B 20.2 2 6.2 2 220.26.2 cos 80.75 406.2172 ⇒ b 20.15 miles
cos C a2 b 2 c 2
2ab
20.22 406.2172 6.2 2
220.220.15
A 180 17.68 80.75 81.57 81 34
9. Given:
sin B b sin A
a
a 6, b 8, c 12
cos A b2 c 2 a 2
2bc
8 sin 26.4
6
64 144 36
2812
C 180 26.4 36.3 117.3
0.5928 ⇒ B 36.3
0.9528 ⇒ C 17.68 17 41
0.8958 ⇒ A 26.4
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Section 6.2 Law of Cosines 475
10. Given: a 9, b 3, c 11
cos A b2 c 2 a 2
2bc
cos C a2 b 2 c 2
2ab
B 180 A C 12.9
11. Given:
cos B a2 c 2 b 2
2ac
32 11 2 9 2
2311
92 3 2 11 2
293
A 50, b 15, c 30
546.49 900 225
223.430
C 180 A B 180 50 29.5 100.6
0.7424 ⇒ A 42.1
0.574 ⇒ C 125.0
a 2 b 2 c 2 2bc cos A 225 900 21530 cos 50 546.49
0.8708 ⇒ B 29.4
⇒ a 23.38
12.
C 108, a 10, b 7
c 2 a 2 b 2 2ab cos C 10 2 7 2 2107 cos 108 192.2624 ⇒ c 13.9
sin B sin C sin 108
b 7 0.4789 ⇒ B 28.7
c 13.9
A 180 108 28.7 43.3
13. Given:
a 9, b 12, c 15
cos C a2 b 2 c 2
2ab
sin A 9 15 3 5 ⇒ A 36.9
B 180 90 36.9 53.1
81 144 225
2912
0 ⇒ C 90
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14. Given:
cos C a2 b 2 c 2
2ab
cos B a2 c 2 b 2
2ac
A 180 B C 20.0
15. Given:
sin B b sin A
a
C B 38.2
a 45, b 30, c 72
452 30 2 72 2
24530
452 72 2 30 2
24572
a 75.4, b 48, c 48
cos A b2 c 2 a 2
2bc
482 48 2 75.4 2
24848
48 sin 103.5
75.4
0.8367 ⇒ C 146.8
0.9736 ⇒ B 13.2
0.2338 ⇒ A 103.5
0.6190 ⇒ B 38.2
(Because of roundoff error, A B C 180. )

476 Chapter 6 Additional Topics in Trigonometry
16. Given:
a 1.42, b 0.75, c 1.25
cos A b2 c 2 a 2
2bc
cos B a2 c 2 b 2
2ac
180 86.7 31.8 61.5
0.752 1.25 2 1.42 2
20.751.25
1.422 1.25 2 0.75 2
21.421.25
0.05792 ⇒ A 86.7
0.8497 ⇒ B 31.8
17. Given:
b 2 a 2 c 2 2ac cos B 26 2 18 2 22618 cos8.25 73.6863
sin C c sin B
b
B 8 15 8.25, a 26, c 18
18 sin8.25
8.58
0.3 ⇒ C 17.51 17 31
A 180 B C 180 8.25 17.51 154.24 154 14
⇒ b 8.58
18. Given:
b 2 a 2 c 2 2ac cos B 140.8268 ⇒ b 11.87
sin A a sin B
b
B 10 35 10.583, a 40, c 30
0.6189 ⇒ A 141.75 141 45
C 180 A B 27.67 or 27 40
19. Given:
21.
22.
b 2 a 2 c 2 2ac cos B
6.2 2 9.5 2 26.29.5 cos75.33
98.86
b 9.94
sin C c sin B
b
B 75 20 75.33, a 6.2, c 9.5 20. Given:
9.5 sin75.33
9.94
0.9246 ⇒ C 67.6
A 180 B C 37.1
d 2 4 2 8 2 248 cos 30
24.57 ⇒ d 4.96
2 360 2
c 2 4 2 8 2 248 cos 150 135.43
c 11.64
⇒ 150
c 2 25 2 35 2 22535 cos 120 2725 ⇒ c 52.2
2 360 2120 120 ⇒
60
d 2 25 2 35 2 22535 cos 60 975 ⇒ d 31.22
30°
4
φ
8
c 2 a 2 b 2 2ab cos C
6.25 2 2.15 2 26.252.15 cos 15.25
17.7563 ⇒ c 4.21
sin B b sin C
c
C 15 15 15.25, a 6.25, b 2.15
0.1342 ⇒ B 7.71 7 43
A 180 B C 157.04 157 2
8
c
d
4
c
120°
25 25
d
θ
35
35
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Section 6.2 Law of Cosines 477
23.
cos 102 14 2 20 2
21014
24.
cos 402 60 2 80 2
24060
1 4 ⇒
104.5
111.8
2 360 2104.5 151 ⇒ 75.5
2 360 2111.80
c 2 40 2 60 2 24060 cos 75.5 4000
68.2
c 63.25
d 2 10 2 14 2 21014 cos 68.2
60
d 13.86
φ
c
40
10
θ
φ
14
20
d
10
40
θ
60
80
14
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25.
cos 152 12.5 2 10 2
21512.5
cos 152 10 2 12.5 2
21510
180 41.41 55.77 82.82
180 97.18
b 2 12.5 2 10 2 212.510 cos97.18 287.50
b 16.96
sin 10
16.96 sin 0.585 ⇒ 35.8
sin 12.5
16.96 sin 0.731 ⇒
77.2
102.8
27. Given:
s a b c
2
a 12, b 24, c 18
27
Area ss as bs c
271539
104.57 square inches
0.75 ⇒ 41.41 26.
0.5625 ⇒ 55.77
47
b
β ε
ω
10
12.5
15
δ µ
µ
12.5 10
α
ω
b
cos 252 17.5 2 25 2
22517.5
180 110.487
a 2 17.5 2 25 2 217.525 cos 110.487
a 35.18
z 180 2 40.974
cos 252 35.18 2 17.5 2
22535.18
27.772
z 68.7
180 41.741
111.3
28. Given:
69.513
s a b c
2
µ
z
a 25, b 35, c 32
46
Area ss as bs c
46211114
385.70 square meters
a
25
α
ω
25
17.5
α
β
17.5
α
25
25
a

478 Chapter 6 Additional Topics in Trigonometry
29. Given:
s a b c
2
a 5, b 8, c 10
23 2 11.5
Area ss as bs c
11.56.53.51.5
19.81 square units
30.
s a b c
2
195212
14 17 7
2
Area ss as bs c
47.7 square units
19
31. Given:
s a b c
2
a 1.24, b 2.45, c 1.25
2.47
Area ss as bs c
2.471.230.021.22
0.27 square feet
32. Given:
s a b c
2
a 2.4, b 2.75, c 2.25
3.7
Area ss as bs c
3.71.30.951.45
2.57 square centimeters
33. Given:
s a b c
2
a 3.5, b 10.2, c 9
11.35
Area ss as bs c
11.357.851.152.35
15.52 square units
34. Given:
s
a 75.4, b 52, c 52
75.4 52 52
2
89.7
Area ss as bs c
89.714.337.737.7
1350 square units
35. Given:
37.
s a b c
2
a 10.59, b 6.65, c 12.31 36.
14.775
Area ss as bs c
14.7754.1858.1252.465
35.19 square units
B 105 32 137
b 2 a 2 c 2 2ac cos B
648 2 810 2 2648810 cos137
1,843,749.862
b 1357.8 miles
From the Law of Sines:
a
sin A b
sin B ⇒ sin A a 648
sin B sin137 0.32548
b 1357.8
s a b c
2
Area ss as bs c
4.650.22.81.65
2.07 square units
⇒ A 19 ⇒ Bearing S 56 W (or 236 for airplane navigation)
W
75°
N
S
Franklin
E
Rosemont
648 mi
32°
Centerville
810 mi
4.45 1.85 3.00
2
4.65
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Section 6.2 Law of Cosines 479
38. Angle at B 180 80 100
b 2 240 2 380 2 2240380 cos 100
B
80°
240 380
233,673.4 ⇒ b 483.4 meters
C
b
A
39.
cos B 1500 2 3600 2 2800 2
0.6824 ⇒ B 46.97
215003600
Bearing at B: 90 46.97: N 43.03 E
cos C 1500 2 2800 2 3600 2
0.3417 ⇒ C 109.98
215002800
1500 meters
B
N
W E
C
S
2800 meters
3600 meters A
Bearing at C:
C 90 46.97 66.95
S 66.95 E
40.
cos 22 3 2 4.5 2
223
127.2
0.60417 41.
C 180 53 67 60
c 2 a 2 b 2 2ab cos C
36 2 48 2 236480.5 1872
c 43.3 mi
N
36 mi
W
c
53°
60°
E
48 mi
67°
S
42. The angles at the base of the tower are 96 and 84. The longer guy wire is given by:
g 12 75 2 100 2 275100 cos 96 17,192.9 ⇒ g 1 131.1 feet
The shorter guy wire
g 2
is given by: g 22 75 2 100 2 275100 cos 84 14,057.1 ⇒ g 2 118.6 feet
g 1
© Houghton Mifflin Company. All rights reserved.
43.
45.
RS 8 2 10 2 164 241 12.8 feet 44.
PQ 1 2 162 10 2 1 356 89 9.4 feet
2
tan P 10
16
P arctan 5 8 32.0
QS 8 2 9.4 2 289.4 cos 32
24.81 5.0 feet
s a b c
2
145 257 290
2
Area ss as bs c
346
3462018956 18,617.7 square feet
A 180 40 20 120
x
sin 20
sin 120 10 40°
10
3.95 feet
C
B
20°
50°
x
46. The height is h 70 sin 70 65.778.
Area base height
10065.778 6577.8 square meters
70°
A

480 Chapter 6 Additional Topics in Trigonometry
47. (a)
(c)
7 2 1.5 2 x 2 21.5x cos
49 2.25 x 2 3x cos
10
0 2
0
0
.
(d) Note that x 8.5 when and
and x 5.5 when
Thus, the distance is
2,
28.5 5.5 23 6 inches.
(b)
x 2 3x cos 46.75
x 2 3x cos
3 cos
2 2 46.75
3 cos
2 2
x 3 cos
2 2 187
4 9 cos2
4
x 3 cos
±
2
187 9 cos 2
4
Choosing the positive values of x, we have
x 1 2 3 cos 9 cos 2 187.
48. (a)
(b)
(c)
(d)
d 2 10 2 7 2 2107 cos ⇒ d 149 140 cos
arccos
10 2 7 2 d 2
s 360
360
2107
2r 360
45
arccos
149 d2
140
d (inches) 9 10 12 13 14 15 16
(degrees) 60.9 69.5 88.0 98.2 109.6 122.9 139.8
s (inches) 20.88 20.28 18.99 18.28 17.48 16.55 15.37
49. False. This is not a triangle! 5 10 < 16 50. True. The third side is found by the Law of Cosines.
The other angles are determined by the Law of Sines.
51. False. s a b c , not a b c .
2
3
52.
54.
1
2 bc1 cos A 1 2 bc 1 b2 c 2 a 2
2bc
53.
cos A
a
cos B
b
1 2 bc 2bc b2 c 2 a 2
2bc
1 4 b c2 a 2
1 b c ab c a
4
b c a
2
a b c
2
cos C
c
b c a
2
a b c
2
b2 c 2 a 2
a2bc
a2 b 2 c 2
2abc
a2 c 2 b 2
b2ac
1
2 bc1 cos A 1 2 bc 1 b2 c 2 a 2
2bc
a2 b 2 c 2
c2ab
1 2 bc 2bc b2 c 2 a 2
2bc
1 4 a2 b c 2
1 a b ca b c
4
a b c
2 a b c
2
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55. Since 0 < C < 180, cos
C
On the other hand,
2 1 cos C
2
Hence, cos
C
2 1 a2 b 2 c 2 2ab
2
ss c 1 2 a b c 1 2 a b c c
.
2ab a 2 b 2 c 2
.
4ab
Section 6.2 Law of Cosines 481
1 2 a b c1 a b c
2
1 4 a b2 c 2
Thus,
1 4 a2 b 2 2ab c 2 .
ss c
ab
a 2 b 2 2ab c 2
4ab
56. Since 0 < C < 180, sin
C
Hence,
On the other hand,
2 1 cos C
2
sin
C
2 1 a2 b 2 c 2 2ab
2
s as b
1
2 a b c a 1 2 a b c b
1 2 b c a1 a c b
2
.
and we have verified that cos
C
2
2ab a 2 b 2 c 2
.
4ab
ss c
.
ab
1 c a bc a b
4
1 4 c2 a b 2
© Houghton Mifflin Company. All rights reserved.
Thus,
s as b
ab
57. Given:
a 2 b 2 c 2 2bc cos A
1 4 c2 a 2 b 2 2ab.
12 2 30 2 c 2 230c cos 20
c 2 60 cos 20c 756 0
c 2 a 2 b 2 2ab
4ab
Solving this quadratic equation, c 21.97, 34.41.
—CONTINUED—
a 12, b 30, A 20
sin
C
2 .

482 Chapter 6 Additional Topics in Trigonometry
57. —CONTINUED—
For c 21.97,
For c 34.41,
Using the Law of Sines,
For
For
cos B a2 c 2 b 2
2ac
cos B a2 c 2 b 2
2ac
122 21.97 2 30 2
21221.97
C 180 121.2 20 38.8
122 34.41 2 30 2
21234.41
C 180 58.8 20 101.2.
sin B b sin A
a
⇒
B 58.8, C 180 58.8 20 101.2
c
B 121.2, C 180 121.2 20 38.8
30 sin 20
12
0.5184 ⇒ B 121.2
0.5183 ⇒ B 58.8
and
0.8551
B 58.8 or 121.2.
a sin C
sin A 34.42.
and c a sin C
sin A 21.98.
58. Answers will vary. The Law of Cosines can be used to solve the single-solution
case of SSA. It cannot be used for the no-solution case.
59. arcsin1 because sin
2
2 1.
61. 3 because tan 3 tan1
3
3.
60. cos1 0 because cos
2
2 0.
62. arcsin because sin
3
3 3
2
3
2 .
Section 6.3
Vectors in the Plane
1u u, 0u 0
■ A vector v is the collection of all directed line segments that are equivalent to a given directed line
segment PQ \ .
■ You should be able to geometrically perform the operations of vector addition and scalar multiplication.
■ The component form of the vector with initial point P p 1 , p 2 and terminal point Q q 1 , q 2 is
PQ \ q 1 p 1 , q 2 p 2 v 1 , v 2 v.
■ The magnitude of v v 1 , v 2 is given by v v 12 v 22 .
■ You should be able to perform the operations of scalar multiplication and vector addition in component form.
■ You should know the following properties of vector addition and scalar multiplication.
(a) u v v u
(b) u v w u v w
(c) u 0 u
(d) u u 0
(e) cdu cdu
(f) c du cu du
(g) cu v cu cv
(h)
(i) cv c v
© Houghton Mifflin Company. All rights reserved.
—CONTINUED—

Section 6.3 Vectors in the Plane 483
Section 6.3
—CONTINUED—
■ A unit vector in the direction of v is given by
■ The standard unit vectors are i 1, 0 and j 0, 1. v v 1 , v 2 can be written as v v 1 i v 2 j.
■ A vector v with magnitude v and direction can be written as v ai bj vcos i vsin j
where tan ba.
u v
v .
Vocabulary Check
1. directed line segment 2. initial, terminal 3. magnitude
4. vector 5. standard position 6. unit vector
7. multiplication, addition 8. resultant 9. linear combination, horizontal, vertical
1. u 6 2, 5 4 4, 1 v 2.
u 3 0, 4 4 3, 8
v 0 3, 5 3 3, 8
u v
3. Initial point: 0, 0
Terminal point: 4, 3
v 4 0, 3 0 4, 3
v 4 2 3 2 5
4. Initial point: 0, 0
Terminal point: 4, 2
v 4 0, 2 0 4, 2
v 4 2 2 2 20 25 4.47
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5. Initial point:
Terminal point:
2, 2
1, 4
v 1 2, 4 2 3, 2
v 3 2 2 2 13 3.61
7. Initial point:
Terminal point:
v 3 3, 3 2 0, 5
v 5
9. Initial point:
Terminal point:
3, 2
2 5 , 1
3, 3
1, 2 5
v 1 2 5 , 2 5 1 3 5 , 3 5
v 3 5 2 3 5 2 18
25 3 5 2
6. Initial point: 1, 1
Terminal point: 3, 5
v 3 1, 5 1 4, 6
v 4 2 6 2 52 213 7.21
8. Initial point: 4, 1
Terminal point: 3, 1
v 3 4, 1 1 7, 0
v 7 2 0 2 7
10. Initial point:
Terminal point:
7 2 , 0
0, 7 2
v 0 7 2 , 7 2 0 7 2 , 7 2
v 7 2 2 7 2 2 98 4 7 2 2

484 Chapter 6 Additional Topics in Trigonometry
11. Initial point:
Terminal point:
2
3 , 1 12. Initial point:
1 2 , 4 5
v 1 2 2 3 , 4 5 1 7 6 ,
9
5
7
v 6 2 9
5 2 4141 2.1450
30
Terminal point:
5 2 , 2
1, 2 5
v 1 5 2 , 2 5 2 3 2 ,
v 3 2 2
12
5 2
389
10
12
5
2.8302
13.
v 14.
3u 15.
u v 16.
u v
y
y
y
y
u
v
x
3u
u + v
v
u − v
x
−v
u
x
u
x
−v
17.
u 2v 18.
y
20.
y
y
y
v
u + 2v
2v
v
v 1 2 u 19. x
v − 1 u
2
u
−3v
v
u
x
− 1 2 u
x
2u
u
x
21.
24.
y
u
y
2u
2v
u + 2v
x
3v
2u + 3v
x
22.
y
23.
1 v v
2
u
25.
x
1
− 2
u
y
v − 1 u 2
u 4, 2, v 7, 1
(a) u v 11, 3
(b) u v 3, 1
(c) 2u 3v 8, 4 21, 3 13, 1
(d) v 4u 7, 1 16, 8 23, 9
v
x
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Section 6.3 Vectors in the Plane 485
26. (a) u v 5, 3 4, 0 1, 3
(b) u v 5, 3 4, 0 9, 3
(c) 2u 3v 25, 3 34, 0 22, 6
(d) v 4u 4, 0 45, 3 16, 12
27.
u 6, 8, v 2, 4
(a) u v 4, 4
(b) u v 8, 12
(c) 2u 3v 12, 16 6, 12 18, 28
(d) v 4u 2, 4 46, 8 22, 28
28. (a) u v 0, 5 3, 9 3, 4
(b) u v 3, 14
(c) 2u 3v 0, 10 9, 27 9, 37
(d) v 4u 3, 9 0, 20 3, 11
29.
u i j, v 2i 3j
(a) u v 3i 2j
(b) u v i 4j
(c) 2u 3v 2i 2j 6i 9j 4i 11j
(d) v 4u 2i 3j 4i 4j 6i j
30.
u 2i j, v i j
(a) u v i
(b) u v 3i 2j
(c) 2u 3v 4i 2j 3i 3j 7i 5j
(d) v 4u 7i 3j
31. w u v 32. v w u 33. u w v 34.
2v 2w u
2w 2u
© Houghton Mifflin Company. All rights reserved.
35.
38.
40.
6, 0 6 36.
Unit vector: 1 6, 0 1, 0
6
v 3, 4 39.
v 3 2 4 2 5
Unit vector:
v 8, 20
1
5 3, 4 3 5 , 4 5
u 0, 2 37.
u 2
Unit vector:
1
0, 2 0, 1
2
v 1, 1
v 2
Unit vector 1
v v
1 2 , 1
2
2 2 , 2
2
v 24, 7 24 2 7 2 25
1
Unit vector: 2524, 7 24 25 , 25
7
u 1
v v 1
1
8, 20
64 400 29 2, 5 2
29 , 5
29 229
29 , 529
29
41. u 1 1
4i 3j 1 4i 3j 4 5 i 3 v v 16 9 5 5 j

486 Chapter 6 Additional Topics in Trigonometry
42.
w i 2j
u 1
w w
1
1 2 22i 2j
1 i 2j
5
5
5 i 25 j
5
43. u 1 22j j 44.
w 3i
w 3
u 1 33i i
45.
8
1
u u 8
1
5 2 6 5, 6 1
46. v 3 47.
2 u u
8 5, 6
61
4061 , 4861
61 61
3
1
4 2 4 24, 4
3
1
42 4, 4
3 2 , 3 2
32
2 , 32
2
7
1
u u 7
1
3 2 4 3, 4 2
7 3, 4
5
21
5 , 28 5
21
5 i 28 5 j
49. 8
1
48. v 10
1
u u 50.
10
1
4 9 2, 3
20
13 , 30
13
20
13 i 30
13 j
2013
13
i 3013 j
13
51. v 4 3, 5 1 7, 4 7i 4j
53. v 2 1, 3 5 3, 8 3i 8j
55.
3 22i j 3i 3 2 j
3, 3 2
2 3 , 4
v 3 2 u 56. 3
y
1
−1
−2
y
x
1 2 3
u
3
2 u
u u 8 1 2 2, 0
42, 0
8, 0
8i
v 4
1
u u
4
1
5 0, 5
0, 4 4j
52. 3 0, 6 2 3, 8 3i 8j
54. 0 6, 1 4 6, 3 6i 3j
v u 2w
x
2 31, 2
4i 3j 4, 3
y
3
2
w
4
3
2w
u + 2w
2
1 2
w
3
1
x
−1 1 2 3
v 2 3 w 57. 3 4 5
−1
−1
u
2i j 2i 2j
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Section 6.3 Vectors in the Plane 487
58.
v u w 59.
v 1 23u w 60.
v 2u 2w
2i j i 2j
i 3j 1, 3
1 232, 1 1, 2
7 2 , 1 2
4, 2 2, 4
2, 6
y
y
y
−u + w 3
2
w
−u
−2 −1 1
x
2
1
2 w 1
x
−1
1
4
(3u + w)
−1
2
3
−2
2 u
1
−4 −3 −2 −1 2 3 4
2u
−3
−2w
−4
−6
2u − 2w
−7
x
61.
v 5cos 30i sin 30j 62.
v 5,
30
v 8cos 135i sin 135j
v 8,
135
63.
v 6i 6j
64.
v 4i 7j, Quadrant III
v 6 2 6 2 72 62
v 4 2 7 2 65
tan 6 6 1
tan 7
4 7 4 ⇒
240.3
Since v lies in Quadrant IV,
315.
65.
v 2i 5j
v 2 2 5 2 29
tan 5 2
Since v lies in Quadrant II,
111.8.
66. v 12i 15j, Quadrant I
v 12 2 15 2 369 341
tan 15
12 ⇒
51.3
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67.
69.
v 3 cos 0, 3 sin 0
3, 0
4
3
2
1
−4 −3 −2 −1 1 2 3 4
−2
−3
−4
( )
3 6 3 2
− ,
2 2
2
y
θ = 0° (3, 0)
θ = 150°
− 4 −3 −2 −1 1 2 3 4
−2
−3
− 4
x
68.
v 32 cos 150, 32 sin 150 70.
36 2 , 32
2
y
x
v cos 45, sin 45
2
2 , 2
2
v 43 cos 90, sin 90 0, 43
7
6
5
4
3
2
1
−4 −3 −2 −1 1 2 3 4
−1
y
(0, 4 3
θ = 90°
(
x
−2
−1
2
1
−1
−2
y
( )
2 2
,
2 2
θ = 45°
1 2
x

488 Chapter 6 Additional Topics in Trigonometry
71.
v 2
1
y
3 2 1
i 3j 72.
2
4
10 3 10
3 ( , )
2
10 i 3j 5 5
2
1 θ ≈ 71.57°
10
5 i 310 j
5
10
5 , 310
5
−4 −3 −2 −1 1 2 3 4
−2
−3
−4
x
1
v 3
3 2 4
3i 4j
2
3 1
3i 4j ≈ 53.13°
5
9 5 i 12
5 j 9 5 , 12 5
4
3
2
( )
9
,
12
5 5
θ
− 4 −3 −2 −1 1 2 3 4
−2
−3
− 4
y
x
tan 3 1 ⇒
71.57
73.
u 5 cos 60, 5 sin 60 5 2 , 53
2 74.
v 5 cos 90, 5 sin 90 0, 5
u v 5 2 , 53
2 0, 5 5 2 , 5 5 2 3
u 2 cos 30, 2 sin 30 3, 1
v 2 cos 90, 2 sin 90 0, 2
u v 3, 3
75.
u 20 cos 45, 20 sin 45 102, 102 76.
v 50 cos 150, 50 sin 150 253, 25
u v 102 253, 102 25
u 35 cos 25, 35 sin 25 31.72, 14.79
v 50 cos 120, 50 sin 120 25, 253
u v 6.72, 58.09
77.
v i j
y
w 2i j
u v w i 3j
v 2
w 22
v w 10
−1
1
−1
−2
v
α
w
u
1 2
x
78.
cos v2 w 2 v w 2
2v w
90
v 3i j
w 2i j
u v w i 2j
cos v2 w 2 v w 2
2v w
45
2 8 10
22 22 0
10 5 5
2105
2
2
y
3
2
1 θ
v
u
−3 −2 −1
−1
w 3
−2
−3
x
© Houghton Mifflin Company. All rights reserved.

Section 6.3 Vectors in the Plane 489
79.
u 400 cos 25i 400 sin 25j
v 300 cos 70i 300 sin 70j
u v 465.13i 450.96j
u v 465.13 2 450.96 2 647.85
arctan
450.96
465.13 44.11
500
400
300
200
100
y
44.11°
−100 100 200 300 400 500
−100
647.85
x
80. Analytically: v 400cos 25, sin 25
u 300cos 135, sin 135
u v 150.39, 381.18
u v 409.77
arctan
381.18
150.39 68.47
600
400
200
−400 −200
200 400
y
409.77
68.47°
x
81. Force One:
Force Two:
Resultant Force:
u 45i
v 60 cos i 60 sin j
u v 45 60 cos i 60 sin j
u v 45 60 cos 2 60 sin 2 90
2025 5400 cos 3600 8100
5400 cos 2475
cos 2475
5400 0.4583
62.7
© Houghton Mifflin Company. All rights reserved.
82. Force One: u 3000i
Force Two: v 1000 cos i 1000 sin j
Resultant Force: u v 3000 1000 cos i 1000 sin j
u v 3000 1000 cos 2 1000 sin 2 3750
9,000,000 6,000,000 cos 1,000,000 14,062, 500
6,000,000 cos 4,062, 500
cos 4,062,500
6,000,000 0.6771
47.4
83. Horizontal component of velocity: 70 cos 40 53.62 ftsec
Vertical component of velocity: 70 sin 40 45.0 ftsec
84. Horizontal component of velocity: 1200 cos 4 1197.1 ftsec
Vertical component of velocity: 1200 sin 4 83.7 ftsec

490 Chapter 6 Additional Topics in Trigonometry
85. Rope AC \
: u 10i 24j
The vector lies in Quadrant IV and its reference angle is arctan 12 5 .
u u cosarctan 12 5 i sinarctan 12 5 j
Rope BC \
: v 20i 24j
The vector lies in Quadrant III and its reference angle is arctan 6 5.
v v cosarctan 6 5 i sinarctan 6 5j
Resultant: u v 5000j
u cosarctan 12 5 v cosarctan 6 5 0
u sinarctan 12 5 v sinarctan 6 5 5000
Solving this system of equations yields:
T AC u 3611.1 pounds
T BC v 2169.5 pounds
86. Left cable: u ucos 155.7i sin 155.7j
Right cable: v vcos 44.5i sin 44.5j
u v 20,240j ⇒ u cos 155.7 v cos 44.5 0
u sin 155.7 v sin 44.5 20,240
⇒ 0.9114u 0.7133v 0
0.4115u 0.7009v 20,240
Solving this system for u and v yields:
Tension of left cable: u 15,485 pounds
Tension of right cable: v 19,786 pounds
87. (a) Tow line 1:
(b)
Tow line 2:
Resultant: u v 6000i u cos u cosi ⇒
Domain:
T
10
0 ≤
u ucos i sin j
v ucosi sinj
T u 3000 sec
20
< 90
6000 2u cos ⇒ u 3000 sec
30
40
50
60
3046.3 3192.5 3464.1 3916.2 4667.2 6000.0
(d) The tension increases because the component in the direction
of the motion of the barge decreases.
(c)
7500
0 90
0
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Section 6.3 Vectors in the Plane 491
88. (a)
(b)
(c)
u ucos90 i sin90 j
v vcos90 i sin90 j
u v 100j ⇒ u cos90 v cos90 0
100 u sin90 u sin90
Hence,
Domain: 0 ≤
120
2u cos
u 50 50 sec .
cos
⇒ u v , and
< 90
10 20 30 40 50 60
T 50.8 53.2 57.7 65.3 77.8 100
0 90
0
(d) The vertical component of the vectors decreases as
89. Airspeed:
Groundspeed:
w v u
w 58.50 2 116.49 2 130.35 kmhr
arctan
116.49
Direction: N 26.7 E
v 860cos 302i sin 302j
u 800cos 310i sin 310j
58.50 63.3
increases.
w u v 800cos 310i sin 310j 860cos 302i sin 302j 58.50i 116.49j
© Houghton Mifflin Company. All rights reserved.
90. (a)
580 mph
y
N
W
S
28° 45°
60 mph
E
x
(b)
(c)
(d)
(e)
w 60cos 45, sin 45 302, 302
v 580cos 118, sin 118 272.3, 512.1
w v 229.9, 554.5
w v 600.3 mph
tan 554.5
229.9 ⇒ 112.5
Direction: N 22.5 W (or 337.5 using airplane navigation)

492 Chapter 6 Additional Topics in Trigonometry
91. (a)
(b)
(c)
u 220i, v 150 cos 30i 150 sin 30j
u v 220 753i 75j
u v 220 753 2 75 2 357.85 newtons
tan
75
220 753 ⇒
u v 220i 150 cos i 150 sin j
M u v 220 2 150 2 cos 2 sin 2 2220150 cos
70,900 66,000 cos 10709 660 cos
arctan
15 sin
M
0
22 15 cos
30
60
12.1
90
120
150
180
370.0 357.9 322.3 266.3 194.7 117.2 70.0
0 12.1 23.8 34.3 41.9 39.8 0
(d)
500
45
(e) For increasing , the two vectors tend to work
against each other resulting in a decrease in the
magnitude of the resultant.
0 180
0
0 180
0
92. (a)
u w T 0 ⇒ u T cos90 0
u T sin 0
1 T sin90 0
1 T cos 0 ⇒
Hence, u T sin sec sin tan , 0 ≤ < 2.
(b)
0 10 20 30 40 50 60
T 1 1.02 1.06 1.15 1.31 1.56 2
u 0 0.18 0.36 0.58 0.84 1.19 1.73
T T cos90 i sin90 j
(d) Both T and u increase as increases, and approach each
other in magnitude.
T sec , 0 ≤
<
2
(c)
2
θ
T
1 lb
0 60
0
93. True. See page 424. 94. True, u v . 95. True. In fact, a b 0.
v
96. True 97. True. a and d are parallel, and 98. True. c and s are parallel, and
pointing in opposite directions. pointing in the same direction.
T
⏐⏐
u⏐
u
© Houghton Mifflin Company. All rights reserved.

Section 6.3 Vectors in the Plane 493
99. True 100. False. In fact, v s w. 101. True. a w 2a 2d
102. True. a d a a 0 103. False. u v 2u and
2b t 22u 4u
104. True
t w s and b a c s
105. (a) The angle between them is 0.
(b) The angle between them is 180.
(c) No. At most it can be equal to the sum when
the angle between them is 0.
106.
F 1 10, 0, F 2 5cos , sin
(a) F 1 F 2 10 5 cos , 5 sin
F 1 F 2 10 5 cos 2 5 sin 2
100 100 cos 25 cos 2 25 sin 2
54 4 cos cos 2 sin 2
54 4 cos 1
55 4 cos
(c) Range: 5, 15
Maximum is 15 when
0.
Minimum is 5 when .
(b)
20
0 2
0
(d) The magnitude of the resultant is never 0
because the magnitudes of F 1 and F 2 are
not the same.
© Houghton Mifflin Company. All rights reserved.
107. Let v cos i sin j.108. The following program is written for a TI-82 or
TI-83 graphing calculator. The program sketches
v cos 2 sin 2 1 1
two vectors u ai bj and v ci dj in
Therefore, v is a unit vector for any value of .
standard position, and then sketches the vector
difference u v using the parallelogram law.
PROGRAM: SUBVECT
:Input “ENTER A”, A
:Input “ENTER B”, B
:Input “ENTER C”, C
:Input “ENTER D”, D
:Line (0, 0, A, B)
:Line (0, 0, C, D)
:Pause
:A-C→E
:B-D→F
:Line (A, B, C, D)
:Line (A, B, E, F)
:Line (0, 0, E, F)
:Pause
:ClrDraw
:Stop
109.
u 5 1, 2 6 4, 4 110.
v 9 4, 4 5 5, 1
u v 1, 3
v u 1, 3
u 80 10, 80 60 70, 20
v 20 100, 70 0 80, 70
u v 70 80, 20 70 10, 50
v u 80 70, 70 20 10, 50

494 Chapter 6 Additional Topics in Trigonometry
111.
6x 4
y
7y 214x1 5 12x 4 y 5 y 2
x
12x 3 y 7 , x 0, y 0
112. 5s 5 t 5
3s 2 3s3
50t1
10t 4, s 0
113.
18x 0 4xy 2 3x 1 16x2 y 2 3
x
114.
5ab 2 a 3 b 0 2a 0 b 2 5ab 2 1
a 3 1
2b 2
48xy 2 , x 0
5
4a 2, b 0
115. 2.1 10 9 3.4 10 4 7.14 10 5
116. 6.5 10 6 3.8 10 4 24.7 10 10 2.47 10 11 118.
117.
sin x 7 ⇒ 49 x2 7 cos
x 2 49 7 tan
7
x
x
x 2 − 49
θ
49 − x 2
θ
7
119.
cot x
10
⇒ x 2 100 10 csc
120.
x 2 4 2 cot
100 + x 2 10
x
2
θ
x
θ
x 2 − 4
121.
cos xcos x 1 0 122.
sin x2 sin x 2 0
123.
cos x 0 ⇒ x
2 n
cos x 1 ⇒ x 2n
3 sec x 4 10 124.
sec x 2
cos x 1 2
x
3
2n,
5
3 2n
sin x 0 or sin x 2
2
x n or x 5
4 2n
cos x cot x cos x 0
cos xcot x 1 0
x 7
4 2n
cos x 0 or cot x 1
x
2 n or x
4 n
© Houghton Mifflin Company. All rights reserved.

Section 6.4 Vectors and Dot Products 495
Section 6.4
Vectors and Dot Products
■ Know the definition of the dot product of u u 1 , u 2 and v v 1 , v 2 .
u v u 1 v 1 u 2 v 2
■ Know the following properties of the dot product:
1. u v v u
2. 0 v 0
3. u v w u v u w
4. v v v 2
5. cu v cu v u cv
■ If is the angle between two nonzero vectors u and v, then
w
■ The vectors u and v are orthogonal if u v 0.
■ Know the definition of vector components. u w 1 w 2 where w 1 and w 2 are orthogonal, and is
parallel to v. w 1 is called the projection of u onto v and is denoted by
1
■
cos u v
u v .
w 1 proj v u
u v
v 2 v.
Then we have w 2 u w 1 .
Know the definition of work.
\
1. Projection form: W proj PQ
F PQ \
2. Dot product form: W F PQ \
© Houghton Mifflin Company. All rights reserved.
Vocabulary Check
1. dot product 2.
u v
u v
3. orthogonal
u v
\
4. 5. proj PQ
F PQ \ ,
v F 2 v
PQ\
1. u v 6, 3 2, 4 62 34 0 2. u v 4, 1 2, 3
42 13 11
3. u v 5, 1 3, 1 53 11 14 4. u v 3, 2 2, 1 32 21 4
5. u 2, 2
6.
u u 22 22 8, scalar
7. u 2, 2, v 3, 4 8.
u 2v 2u v 43 44 4, scalar
v w 3, 4 1, 4
31 44 19, scalar
4u v 42, 2 3, 4
46 8
8, scalar

496 Chapter 6 Additional Topics in Trigonometry
9.
3w vu 31, 4 3, 42, 2 10.
33 1242, 2
572, 2
114, 114, vector
u 2vw 2, 2 23, 41, 4
26 281, 4
41, 4
4, 16, vector
11.
u 5, 12 12.
u u u 5 2 12 2 13
u 2, 4
u u u
22 44
20 25
13.
u 20i 25j
u u u 20 2 25 2 1025 541
14.
u 6i 10j
u u u 66 1010 136 234
15.
u 4j 16.
u u u 44 4
u 9i
u u u 99 0 81 9
17.
u 1, 0, v 0, 2 18.
cos u v
u v 0
12 0 ⇒
90
u 4, 4, v 2, 0
cos u v
u v
2
2
135
42 40
422
19.
21.
u 3i 4j, v 2i 3j 20.
cos u v 6 12
u v 513 6
513
arccos
6
u 2i, v 3j
513 70.56
cos u v
u v 0
23 0 ⇒
90
u 2i 3j, v i 2j
cos u v
u v
21 32
2 2 3 2 1 2 2 2
8
65 0.992278
7.13
22. u v 0, 4 3, 0 0 ⇒
90
© Houghton Mifflin Company. All rights reserved.

Section 6.4 Vectors and Dot Products 497
23.
u cos
v cos 3
4 i 3
sin 4 j 2 2 i 2
2 j
u v 1
3 i sin
cos u v
u v u v 1 2 2 2 3
2 2 2 6
2
4
arccos
2 6
3 j 1 2 i 3
2 j
4
75 5
12
24.
u cos
4 i sin
v cos
2
3 i sin 2
3 j 1 2 i 3
2 j
cos u v
uv
4 j 2
2 i 2
2 j
24 64
11
6 2
4
⇒
75 5
12
25.
u 2i 4j, v 3i 5j 26.
u 6i 3j, v 8i 4j
cos u v
u v 6 20
2034
cos u v 68 34
u v 4580
1
−1
−1
−2
y
13170
170
1
⇒
2 3 4 5 6
4.40
x
36
60 0.6
53.13
v
−10 −8 −6 −4
−2
2 4
u
−4
8
6
4
2
y
x
−3
−6
−4
−5
u
v
−8
−6
© Houghton Mifflin Company. All rights reserved.
27.
u 6i 2j, v 8i 5j 28.
cos u v 48 10
u v 4089
−2
2
−2
−4
−6
y
29890
890
4 6 8 10
u
v
⇒
13.57
x
u 2i 3j, v 4i 3j
cos u v 24 33
0.0555
u v 1325
93.18
4
3
2
1
y
−2 −1
−1
1 2 3 4 5 6
−2
u
−3
v
x
−8
−4
−10

498 Chapter 6 Additional Topics in Trigonometry
29.
P 1, 2, Q 3, 4, R 2, 5
PQ \ 2, 2, PR \ 1, 3, QR \ 1, 1
PR \
cos PQ\
PQ \ PR \ 8
2210
QR \
cos PQ\
PQ \ QR \ 0 ⇒
Thus,
90
180 26.6 90 63.4.
⇒ arccos
2
5 26.6
30.
P 3, 0, Q 2, 2, R 0, 6 31.
PQ \ 5, 2, QR \ 2, 4, PR \ 3, 6,
QP \ 5, 2
PR \
cos PQ\
PR \ PR \ 27
2945 ⇒ 41.6
QP \
cos QR\
QR \ QP \ 2
2029 ⇒ 85.2
180 41.6 85.2 53.2
u v u v cos
936 cos 3
4
324 2
2
1622
32.
u v u v cos
33.
u 12, 30, v 1 2 , 5 4
412 cos
3 48 1 2 24
u 24v ⇒ u and v are parallel.
34.
u 15, 45, v 5, 12 35.
u kv ⇒ Not parallel
u v 0 ⇒ Not orthogonal
Neither
u 1 43i j, v 5i 6j
u kv ⇒ Not parallel
u v 0 ⇒ Not orthogonal
Neither
36. u j, v i 2j 37. u 2i 2j, v i j
u kv ⇒ Not parallel
u v 0 ⇒ Not orthogonal
Neither
u v 0 ⇒ u and v are orthogonal.
38. 4v 42i j 8i 4j u ⇒ parallel 39. u v 2, k 3, 2 6 2k 0 ⇒ k 3
40. u v 3, 2 2, k 6 2k 0 ⇒ k 3
41. u v 1, 4 2k, 5 2k 20 0 ⇒ k 10
42. u v 3k, 5 2, 4 6k 20 0 ⇒ k 10 3
© Houghton Mifflin Company. All rights reserved.
43. u v 3k, 2 6, 0 18k 0 ⇒ k 0
44. u v 4, 4k 0, 3 12k 0 ⇒ k 0

Section 6.4 Vectors and Dot Products 499
45.
u 3, 4, v 8, 2 46.
w 1 proj v u
u v
v 2 v
32
68 v 8 16
8, 2 4, 1
17 17
w 2 u w 1 3, 4 16 13
4, 1 1, 4
17 17
u w 1 w 2 16 13
4, 1 1, 4
17 17
u 4, 2, v 1, 2
u
w 1 proj v u
v
v2 v 01, 2 0, 0
w 2 u w 1 4, 2 0, 0 4, 2
u 4, 2 0, 0
47.
u 0, 3, v 2, 15 48.
u
w 1 proj v u
v 45
v2 v 2, 15
229
w 2 u w 1 0, 3 45 2, 15
229
90
229 , 229 12 6 15, 2
229
u w 1 w 2 45
6
2, 15 15, 2
229 229
u 5, 1, v 1, 1
u
w 1 proj v u
v
v 2 v 4 1, 1 21, 1
2
w 2 u w 1 5, 1 21, 1
u 3, 3 2, 2
3, 3 31, 1
49. proj v u u since they are parallel.
50. Because u and v are parallel, the projection of u
proj v u u v
onto v is u.
v v 2
18 8
36 16 v 26 6, 4 3, 2 u
52
© Houghton Mifflin Company. All rights reserved.
51. proj v u 0 since they are perpendicular.
52. Because u and v are orthogonal, the projection of u
proj v u u v
onto v is 0.
since u v 0.
v v 0, 2
53.
55.
u 2, 6
For v to be orthogonal to u, u v must equal 0.
Two possibilities:
6, 2 and 6, 2
54. For v to be orthogonal to u 7, 5, their dot
product must be zero.
Two possibilities: 5, 7, 5, 7
u 1 2 i 3 4 j 56. u 5 2 i 3j
For v to be orthogonal to u, u v must equal 0.
For v to be orthogonal to u, u v must be equal
Two possibilities: 3 and 3 4 , 1 2 4 , to 0.
1 2
Two possibilities: v 3i 5 and v 3i 5 2 j
2 j

500 Chapter 6 Additional Topics in Trigonometry
57. W proj where PQ \
PQ
\v PQ \ 4, 7 and
58.
v 1, 4
proj PQ
\v
v PQ \
32
PQ \ 4, 7
65 2PQ\
W proj \v PQ \
3265
PQ
65 65 32
P 1, 3, Q 3, 5, v 2i 3j
work proj PQ
\v PQ \ where
PQ \ 4, 2 and v 2, 3.
proj PQ
\v
v PQ \
14
PQ \ 4, 2
20 2PQ\
work proj PQ
\v PQ\
1420
20 20 14
59. (a)
u v 1245, 2600 12.20, 8.50 60.
124512.20 26008.50 37,289
This is the total dollar value of the picture
frames produced.
(b) Multiply v by 1.02.
u 3240, 2450, v 1.75, 1.25
(a) u v 32401.75 24501.25 8732.50
This gives the total revenue earned.
(b) To increase prices by
1.025:
2 1 2
percent, multiply v by
1.0251.75, 1.25, scalar multiplication
61. (a) F 30,000j \ , Gravitational force
(b)
v cosd, sind
w 1 proj v F
F v
v 2v F vv 30,000 sindcos d, sin d
Force needed:
d
0
30,000 sind
1
2
3
30,000 sin d cos d, 30,000 sin 2 d
4
5
Force 0 523.6 1047.0 1570.1 2092.7 2614.7 3135.9 3656.1 4175.2 4693.0 5209.4
6
7
8
9
10
(c)
w 2 F w 1 30,000j 2614.7cos5i sin5j 2604.75i 29,772.11j
w 2 29,885.8 pounds
62. F 5400j, Gravitational force
v cos 10i sin 10j
w 1 proj v F F v v F
v2 vv 937.7v
Magnitude: 937.7 pounds
w 2 F w 1
5400j 937.7cos 10i sin 10j
923.45i 5237.17j
w 2 5318 pounds
64. W 4520 cos 30 779.4 foot-pounds
63. (a)
(b)
F 15,691cos 30, sin 30
PQ \ d1, 0
W F PQ \ 15,691 3
2 d 13,588.8d
d 0 200 400 800
Work 0 2,717,761 5,435,522 10,871,044
© Houghton Mifflin Company. All rights reserved.

Section 6.4 Vectors and Dot Products 501
65.
F 250, PQ \ 100, 3066.
W F PQ \ cos
250100 cos 30
25,000 3
2
12,5003 foot-pounds
21,650.64 foot-pounds
F 25, PQ \ 50, 20
W F PQ \ cos
2550 cos 20
1174.62 foot-pounds
67. W cos 252040 725.05 foot-pounds 68. Work cos 202512 281.9 foot-pounds
20 pounds
25°
40 ft
69. True. u v 0 70. False. Work is a scalar, not a
vector.
71. u v 0 ⇒ they are
orthogonal (unit vectors).
72. (a)
(b)
u v 0 ⇒ u and v are orthogonal and
2 .
u v > 0 ⇒ cos > 0 ⇒ 0 ≤
(c) u v < 0 ⇒ cos < 0 ⇒
2 <
<
≤
2
73. (a) proj v u u ⇒ u and v are parallel.
(b) proj v u 0 ⇒ u and v are orthogonal.
© Houghton Mifflin Company. All rights reserved.
74. Let u and v be two adjacent sides of the rhombus,
u v. The diagonals are u v and u v.
u v u v u u v u u v v v
u 2 v 2 0
Hence, the diagonals are perpendicular.
u + v
v
u − v
u
76. u cv dw cu v du w c0 d0 0
77. From trigonometry, x cos and y sin .
Thus, u x, y cos i sin j.
y
75. Use the Law of Cosines on the triangle:
u v 2 u 2 v 2 2u v cos
u 2 v 2 2u v
v
u − v
θ
u
(x, y)
u
y
θ
x
i
x

502 Chapter 6 Additional Topics in Trigonometry
78. From trigonometry,
y
x cos
2
and
y sin
2 .
j
u
(x, y)
Thus, u x, y cos 2
i sin
2
j.
y
θ
x
x
79. gx f x 4 is a horizontal shift of f four units 80. g is a reflection of f with respect to the x-axis.
to the right.
81. gx f x 6 is a vertical shift of f six units 82. g is a horizontal shrink of f .
upward.
83. 4 1 2i 1 1 2i
84. 8 5 22i 5 5 22i
85. 3i4 5i 12i 15 15 12i
86. 2i1 6i 2i 12 12 2i
87. 1 3i1 3i 1 3i 2 1 9 10 88. 7 4i7 4i 49 16 65
89.
3
1 i 2
2 3i 3
1 i 1 i
1 i 2
2 3i 2 3i
2 3i
3 3i
2
4 6i
13
39 39i 8 12i
26
47
26 27
26 i
90.
92.
6 4 i
4 i 4 i 3
1 i 1 i 24 6i
3 3i
1 i 17 2
−4−3 −2 −1
Imaginary
axis
5
4
3 3i
2
1
−2
−3
−4
−5
1 2 3 4 5
Real
axis
48 12i 51 51i
34
3
34 63
34 i
3i 93.
−4−3 −2 −1
Imaginary
axis
8
7
6
5
4
3
2
1
−2
1 + 8i
91.
1 2 3 4 5
2i
Real
axis
−4−3 −2 −1
Imaginary
axis
5
4
3
2
1
−2
−3
−4
−5
1 8i 94.
1 2 3 4 5
−2i
9 7i
−8−6 −4 −2
Real
axis
Imaginary
axis
10
8
6
4
2
−4
−6
−8
−10
2 4
6 8 10
9 − 7i
Real
axis
© Houghton Mifflin Company. All rights reserved.

Section 6.5 Trigonometric Form of a Complex Number 503
Section 6.5
Trigonometric Form of a Complex Number
■ You should be able to graphically represent complex numbers.
■ The absolute value of the complex numbers z a bi is z a2 b 2 .
■ The trigonometric form of the complex number z a bi is z rcos i sin where
(a) a r cos
(b) b r sin
(c) r a 2 b 2 ; r is called the modulus of z. (d) tan ba; is called the argument of z.
■ Given z and z 2 r 2 cos 2 i sin 1 r 1 cos 1 i sin 1
2:
(a) z 1 z 2 r 1 r 2 cos1 2 i sin1 (b)
z 1
r 1
cos1
z 2 r 2 i sin1 2 0
2
■ You should know DeMoivre’s Theorem: If z rcos i sin , then for any positive integer n,
z n r n cos n i sin n.
■ You should know that for any positive integer n, z rcos i sin has n distinct nth roots given by
nr cos 2k
n
i sin
where k 0, 1, 2, . . . , n 1.
2k
n
Vocabulary Check
1. absolute value 2. trigonometric form, modulus, argument
3. DeMoivre’s 4. nth root
1.
Imaginary
axis
Imaginary
axis
4 42 0 2
16 4
© Houghton Mifflin Company. All rights reserved.
4.
6i 6 2. 5
−4 −3 −2
7
6
5
4
3
2
1
−1
6i
1 2 3 4
Real
axis
7 7 5.
Imaginary
axis
6
4
2
−2
−2
−4
−6
7
2 4 6 8 10
Real
axis
2i 2 3. Real
−4−3 −2 −1
5
4
3
2
1
−2
−3
−4
−5
1 2 3 4
−2i
Real
axis
4 4i 42 4 2 6.
− 4 + 4i
−5 −4 −3 −2 −1
−1
32 42
Imaginary
axis
4
3
2
1
−2
1
Real
axis
−4
−5 −4 −3 −2 −1
−1
−10 − 8
− 6
−5 − 12i
− 4
Imaginary
axis
−2
3
2
1
−2
−3
Imaginary
axis
− 6
− 8
−10
−12
1
2
axis
5 12i 52 12 2
169 13
Real
axis

504 Chapter 6 Additional Topics in Trigonometry
7.
45 35
Imaginary
axis
109
Imaginary
axis
8
10
7
8
6 3 + 6i
6
5
4
3
4
2
Real
2
−8−6 −4 −2 2 8
axis
3 6i 9 36 8. 1
Real
−4 10 − 3i
−4−3 −2 −1 1 2 3 4 5
axis
−6
−8
−2
−10
10 3i 102 3 2 9.
z 3i
r 0 2 3 2 9 3
z 3 cos
2 i sin
2
tan 3 0 , undefined ⇒
2
10.
z 4
11.
z 2 12.
z i
r 4 2 0 2 16 4
tan 0 4 0 ⇒
z 4cos 0 i sin 0
0
r 2 2 0 2 2
tan ⇒
z 2cos i sin
r 0 2 1 2 1
tan undefined ⇒
z cos 3 3
i sin
2 2
3
2
13.
z 2 2i
14.
z 3 3i
r 2 2 2 2 8 22
r 3 2 3 2 32
tan 2 is in Quadrant III.
2 1,
5
4
tan 3 3 1 ⇒
z 32 cos
4 i sin
4
4
z 22 cos 5 5
i sin
4 4
15.
17.
z 3 i 16.
r 3 2 1 2 2
tan 1
3 ⇒
11
z 2 cos 11 11
i sin
6 6
z 5 5i
r 5 2 5 2 50 52
tan 5 5 1 ⇒
z 52 cos 7 7
i sin
4 4
6
7
4
Imaginary
axis
1
−1
−1
−2
−3
−4
−5
2
3
4
5
z 1 3i
r 1 2 3 2 4 2
tan 3
1 3 ⇒
z 2 cos 2 2
i sin
3 3
Real
axis
5 − 5i
2
3
© Houghton Mifflin Company. All rights reserved.

Section 6.5 Trigonometric Form of a Complex Number 505
18.
z 2 2i 19.
z 3 i
r 4 4 22
tan 2 2 1 ⇒
z 22 cos
4 i sin
4
4
r 3 2 1 2 4 2
tan 1 3 3
3
z 2 cos
6 i sin
⇒
6
6
Imaginary
axis
Imaginary
axis
2
−4−3 −2 −1
5
4
3
2
1
−2
−3
−4
−5
2 + 2i
1 2 3 4
5
Real
axis
−1
1
−1
3 + i
1 2
Real
axis
20.
z 1 3i 21.
r 1 3 2
tan 3
1
z 2 cos 4 4
i sin
3 3
Imaginary
axis
⇒
240 4
3
z 21 3i
r 2 2 23 2 16 4
tan 3
1 3 ⇒
z 4 cos 4 4
i sin
3 3
Imaginary
axis
4
3
5
4
3
2
1
−4−3 −2 −1
−1 − 3i
−3
−4
−5
1
2
3
4
5
Real
axis
− 4
−3 −2 −1
−2
−3
−2(1 + 3i) − 4
Real
axis
© Houghton Mifflin Company. All rights reserved.
22.
z 5 3 i
2
5
r 2 3 2 5
2 1 2
100
4
25 5
tan 1
3
Imaginary
axis
2
1
−1
−1
2
3
3
3 4 5
Real
axis
⇒
z 5 cos 11 11
i sin
6 6
11
6
23.
z 8i
r 0 8 2 64 8
tan 8 undefined ⇒
0 ,
z 8 cos 3 3
i sin
2 2
Imaginary
axis
2
1
−4−3 −2 −1 1 2 3 4 5
−2
−3
−4
−5
−6
−7
−8 −8i
Real
axis
3
2
−2
−3
− 4
5 ( 3 − i)
2

506 Chapter 6 Additional Topics in Trigonometry
24.
z 4i
25.
z 7 4i
r 4
r 49 16 65
tan 4 undefined
0
⇒
2
tan 4
7 ⇒
2.62
radians or 150.26
z 4 cos
2 i sin
2
−4−3 −2 −1
Imaginary
axis
5
4 4i
3
2
1
−2
−3
−4
−5
1 2
3
4
5
Real
axis
z 65cos 150.26 i sin 150.26
−7 + 4i
−8
−6
−4
−2
Imaginary
axis
4
2
−2
−4
Real
axis
26.
z 5 i
27.
z 3
r 5 2 1 2 26
r 3 2 0 2 3
tan 1 5 ⇒ 11.3
or 348.7
tan 0 3 0 ⇒
0
z 26cos 11.3 i sin 11.3
Imaginary
axis
z 3cos 0 i sin 0
Imaginary
axis
2
1
−1
−1
−2
−3
−4
1
4
5
5 − i
Real
axis
−4−3 −2 −1
5
4
3
2
1
−2
−3
−4
−5
1
3
2 3
4
5
Real
axis
28.
z 6
29. z 3 3i
r 6
0
z 6cos 0 i sin 0
−8−6 −4 −2
Imaginary
axis
10
8
6
4
2
−4
−6
−8
−10
6
2 4 6 8 10
Real
axis
r 9 3 12 23
tan 3
3
z 23 cos
6 i sin
−4−3 −2 −1
Imaginary
axis
5
4
3
2
1
−2
−3
−4
−5
⇒
3 + 3i
1 2 3 4 5
Real
axis
or 30
6
6
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Section 6.5 Trigonometric Form of a Complex Number 507
30.
z 22 i 31.
r 22 2 1 2 9 3
tan 1
22 2 4
z 3cos19.5 i sin19.5
Imaginary
axis
⇒ 19.5 or 340.5
z 1 2i
r 1 2 2 2 5
tan 2
1 2 ⇒
z 5cos 243.4 i sin 243.4
Imaginary
axis
243.4
1
3
2
−1
−2
2
3
2 2 − i
Real
axis
1
−3 −2 −1
−1 − 2i
−3
1
2
3
Real
axis
32.
z 1 3i 33.
r 1 2 3 2 10
tan 3 1 3 ⇒ 71.6
z 10cos 71.6 i sin 71.6
Imaginary
axis
z 5 2i
r 25 4 29 5.385
tan 2 5 ⇒
Imaginary
axis
21.80
z 29cos 21.80 i sin 21.80
−1
3
2
1
−1
1
1 + 3i
2
3
Real
axis
−4−3 −2 −1
5
4
3
2
1
−2
−3
−4
−5
1
2
3
5 + 2i
4 5
Real
axis
34.
3 i 3.16cos 161.6 i sin 161.6 35.
z 32 7i
3.16cos 2.82 i sin 2.82
r 18 49 67 8.185
© Houghton Mifflin Company. All rights reserved.
−3 + i
−4−3 −2 −1
Imaginary
axis
5
4
3
2
1
−2
−3
−4
−5
1 2
3
4
5
Real
axis
tan 7
32 1.6499 ⇒
z 67cos 301.22 i sin 301.22
−4−3 −2 −1
Imaginary
axis
2
1
−2
−3
−4
−5
−6
−7
−8
1 2 3 4 5
3 2 −7i
Real
axis
301.22

508 Chapter 6 Additional Topics in Trigonometry
36.
8 53i 11.79cos 227.3 i sin 227.3 37.
8 53i 11.79cos 3.97 i sin 3.97
−8−6 −4
Imaginary
axis
10
8
6
4
2
−4
−6
−8
−8 − 5 3i
2 4 6 8 10
Real
axis
2cos 120 i sin 120 2 1 2 3
2 i
−1 + 3i
Imaginary
axis
−4 −3 −2 −1
−1
−2
−3
3
2
1 2
Real
axis
1 3i
38.
5cos 135 i sin 135 5 2 2 i 2
2
39.
52
2
52
2 i
3
2 cos 330 i sin 330 3 2 3
2 1 2 i
33
4
3 4 i
Imaginary
axis
Imaginary
axis
−
5 2
+
5 2
i
2 2
4
2
3
1
2
−2
−1
1 2
Real
axis
−4
−3
−2
−1
1
Real
axis
−1
−2
3 3 −
3 i
4 4
40.
3
4 cos 315 i sin 315 3 4 2
2 i 2 2
41.
Imaginary
axis
32
8
32
8 i
3.75 cos 3 3
i sin
4 4 152 8
−
15 2
+
15 2
i
8 8
Imaginary
axis
3
2
152 i
8
42.
−1
1
−1
−2
−4−3 −2 −1
−2
−3
−4
−5
1
3 2
−
3 2
i
8 8
Imaginary
axis
5
4
3
2
1.5i
1
2
1 2 3 4 5
Real
axis
1.5 cos
2 i sin 2 1.50 i 1.5i 43.
Real
axis
− 4
−3
−4−3 −2 −1
−2
Imaginary
axis
8
7
6
5
4
3
2
1
−2
−1
1
−1
6 cos
3 i sin
3 + 3 3i
1 2 3 4 5
Real
axis
3 6 1 2 3
Real
axis
3 33i
2 i
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Section 6.5 Trigonometric Form of a Complex Number 509
44.
8 cos 5 5
i sin
6 6 8 3 2 i1 45.
2
43 4i
4 cos 3 3
i sin
2 2 40 i 4i
Imaginary
axis
−4 3 + 4i
−7 −6 −5 −4 −3 −2
Imaginary
axis
5
4
3
2
1
−1
−2
−3
1
Real
axis
−2
−1
−1
−2
−3
−4
−4i
1 2
Real
axis
46.
9cos 0 i sin 0 9 47.
Imaginary
axis
3cos18 45 i sin18 45 2.8408 0.9643i
Imaginary
axis
6
2
4
2
−2
−2
2
4
6
8
9
10
Real
axis
1
−1
2.8408 + 0.9643i
1 2 3 4
Real
axis
−4
−6
−2
48.
6cos230 30 i sin230 30 3.8165 4.6297i
Imaginary
axis
−5 −4 −3 −2 −1 1
1
Real
axis
−2
−3
−4
−3.8165 − 4.6297i
−5
© Houghton Mifflin Company. All rights reserved.
49. 5 cos
9 i sin
9
4.6985 1.7101i
51. 9cos 58 i sin 58 4.7693 7.6324i
53.
z 4 = −1
−2 −1
−1
Imaginary
axis
2
2
z 3 = ( −1 + i)
2 z 2 =
−2
i
2
z = (1 + i)
2
Real
1 2
axis
The absolute value of each power is 1.
50. 12 cos 3 3
i sin
5 5 3.71 11.41i
52. 4cos 216.5 i sin 216.5 3.22 2.38i
54.
z 1 21 3i
z 2 1 21 3i 1 21 3i 1 21 3i
z 3 z 2 z 1 21 3i 1 21 3i
1
z 4 z 3 z
1 1 21 3i
1 21 3i
The absolute value of
each is 1.
Imaginary
axis
2
z 2 =
1
( −1 + 3 i)
2
z 3 = −1
−2 −1
z 4 = 1
i
2
( − 1 − 3 )
−2
z =
1
(1 + 3 i)
2
1 2
Real
axis

510 Chapter 6 Additional Topics in Trigonometry
55. 3 cos
3 i sin
3 4 cos
6 i sin
6 34 cos
3
6 i sin
6
3 12 cos
2 i sin
2
56.
3 2 cos
6 i sin
6 6 cos
4 i sin
4 3 2 6 cos
6
4 i sin
9 cos 5 5
i sin
12 12
6
4
57.
58.
59.
5 3cos 140 i sin 140 2 3cos 60 i sin 60 5 3 2 3cos140 60 i sin140 60
10 9 cos 200 i sin 200
1 2cos 115 i sin 115 4 5cos 300 i sin 300 1 2 4 5cos115 300 i sin115 300
2 5cos 415 i sin 415 2 5cos 55 i sin 55
11
20cos 290 i sin 290 2 5 cos 200 i sin 200 11
20 2 5cos290 200 i sin290 200
11
50 cos 490 i sin 490
11
50 cos 130 i sin 130
60.
cos 5 i sin 5cos 20 i sin 20 cos5 20 i sin5 20
cos 25 i sin 25
61.
cos 50 i sin 50
cos50 20 i sin50 20 cos 30 i sin 30
cos 20 i sin 20
5cos4.3 i sin4.3
62. 5 cos2.2 i sin2.2
4cos2.1 i sin2.1 5 cos4.3 2.1 i sin4.3 2.1
4 4
63.
2cos 120 i sin 120
4cos 40 i sin 40
1 2 cos120 40 i sin120 40 1 cos 80 i sin 80
2
64.
65.
66.
cos 7 7
i sin
4 4
cos i sin
cos 3 3
i sin
4 4
18cos 54 i sin 54
6cos54 102 i sin54 102
3cos 102 i sin 102
6cos48 i sin48 6cos 312 i sin 312)
9cos 20 i sin 20
5cos 75 i sin 75 9 cos20 75 i sin20 75
5
9 cos55 i sin55
5
© Houghton Mifflin Company. All rights reserved.
9 cos 305 i sin 305
5

67. (a)
(b)
68. (a)
(b)
69. (a)
(c) 2 2i1 i 2 2 4
3 3i 32 cos 7 7
i sin
4 4
1 i 2 cos 7 7
i sin
4 4
3 3i1 i 32 cos 7 7
i sin
4 4 2
(c) 3 3i1 i 3 3 6i 6i
(b)
2 2i 22 cos 7 7
i sin
4 4
1 i 2 cos
4 i sin
2 2i1 i 22 cos 7 7
i sin
4 4 2 cos
4
2 2i 22cos 45 i sin 45
1 i 2cos45 i sin45
(c) 2 2i1 i 2 2i 2i 2i 2 2 2 4
Section 6.5 Trigonometric Form of a Complex Number 511
7
cos
4
4 i sin 4 4cos 2 i sin 2 4
i sin
7
4 6
7
cos
2
i sin
7
2 6i
2 2i1 i 22cos 45 i sin 452cos45 i sin45 4cos 0 i sin 0 4
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70. (a)
(b)
71. (a)
3 i 2cos 30 i sin 30
1 i 2cos 45 i sin 45
3 i1 i 2cos 30 i sin 302cos 45 i sin 45
22cos 75 i sin 75
6 2
22 4 6 2
4
i
3 1 3 1i
(c) 3 i1 i 3 3 1i i 2 3 1 3 1i
(b)
2i 2cos90 i sin90
1 i 2cos 45 i sin 45
2i1 i 2cos90 i sin90 2cos 45 i sin 45
22cos45 i sin45
22 1 2
1 2 i 2 2i
(c) 2i1 i 2i 2i 2 2i 2 2 2i

512 Chapter 6 Additional Topics in Trigonometry
72. (a)
3 i 3 cos
2 i sin
2
1 i 2 cos
4 i sin
4
(c) 3i1 i 3i 3 3 3i
(b)
3i1 i 3 cos
2 i sin
2 2 cos
32 cos 3 3
i sin
4 4
32 2
2 i2 2
4 i sin
3 3i
4
73. (a)
(b)
2i 2 cos 3 3
i sin
2 2
3 i 2 cos 11 11
i sin
6 6
2i3 i 2 cos 3 3
i sin
2 2 2
4 cos 20 20
i sin
6 6
4 1 2 3
2 i
2 23i
(c) 2i3 i 23i 2
11
cos
6
i sin
11
6
74. (a)
(b)
75. (a)
i cos 3 3
i sin
2 2
1 3i 2 cos
3 i sin
i1 3i cos 3 3
i sin
2 2 2 cos
2 cos 11 11
i sin
6 6
2
3
2 i1 2 3 i
(c) i1 3i i 3
(b)
2 2cos 0 i sin 0
(c) 21 i 2 2i
3
1 i 2 cos 7 7
i sin
4 4
21 i 22 cos 7 7
i sin
4 4
22 2
2 2
2 i
21 i 2 2i
3 i sin
3
76. (a)
(b)
4 4cos i sin
1 i 2 cos
4 i sin
41 i 42 cos 5 5
i sin
4 4
4 4i
(c) 41 i 4 4i
4
42 2
2 2
2 i
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77. (a)
(b)
(c)
78. (a)
(b)
(c)
3 3i 32 cos
4 i sin
1 3i 2 cos 5 5
i sin
3 3
3 3i
1 3i 32
2 cos
4 5
32
2 cos 17
12 i sin 17
12
0.549 2.049i
3 i sin
3 3i
1 3i 1 3i 3 33 3 33i
1 3i 4
2 cos
1.366 0.366i
4
0.549 2.049i
2 2i 22 cos
4 i sin
1 3i 2 cos
3 i sin
2 2i
1 3i 22
2 cos
4
3
3 i sin
12 i sin
1.366 0.366i
Section 6.5 Trigonometric Form of a Complex Number 513
4 5
3
2 2i
1 3i 1 3i 2 23 2 23i
1 3
1 3i 4
2
4
4
12
3
1 3 i
2
© Houghton Mifflin Company. All rights reserved.
79. (a)
(b)
(c)
5 5cos 0 i sin 0 80. (a)
2 2i 22 cos
4 i sin
5
2 2i 5cos 0 i sin 0
22 cos
4 i sin
5
22 cos
22 5 2
2 2
2 i 5 4 5 4 i
5
2 2i 2 2i 52 2i
5 2 2i 4 4 4 5 4 i
4
4
4 i sin 4
(b)
(c)
2 2cos 0 i sin 0
3 i 2 cos 11 11
i sin
6 6
2
3 i 2 2 cos 11
6 i sin 11
6
3
2 1 2 i
2
3 i 3 i 23 i
3
3 i 4 2 1 2 i

514 Chapter 6 Additional Topics in Trigonometry
81. (a)
(b)
(c)
82. (a)
(b)
(c)
4i
1 i
4i 4 cos
2 i sin
4i
1 i 1 i
1 i 4i 4 2 2i
2
4
2 cos
2i 2 cos
2 i sin
1 3i 2 cos 5 5
i sin
3 3
2i
1 3i 2 2 cos 2 5
3 i sin
cos 7
6 i sin 7
6 3 1 2 2 i
2i
1 3i 1 3i 2i 23
3 1 1 3i 4 2 2 i
2
1 i 2 cos 3 3
i sin
4 4
4
2 cos 2 3
4 i sin
2
4 i sin 4
4
2 2
2 2
2 i 2 2i
2 3
2 5
4
3
83. Let z x iy such that:
84. Circle of radius 5
z 2 ⇒ 2 x2 y 2 ⇒ 4 x 2 y 2
Circle with radius of 2
85. Let z x iy.
Circle of radius 4
−6
Imaginary
axis
1
−1
−1
z 4 ⇒ 4 x2 y 2 ⇒ x 2 y 2 16
Imaginary
axis
6
2
−2
−2
1
2 6
Real
axis
Real
axis
z 5,
−6
Imaginary
axis
6
4
2
−2
−2
−4
−6
2 4 6
86. Let z x iy.
Circle of radius 6
Real
axis
z 6 ⇒ 6 x2 y 2 ⇒ x 2 y 2 36
−9
Imaginary
axis
9
3
−3
−3
3 9
Real
axis
© Houghton Mifflin Company. All rights reserved.
−6
−9

Section 6.5 Trigonometric Form of a Complex Number 515
87.
6
Imaginary
axis
88.
4 arctan 1
Imaginary
axis
Let z x iy such that:
Line
tan
6 y x ⇒
θ = π 6
Real
axis
y x
θ = π 4
Real
axis
y
x 1 3 ⇒ y 1 3 x
Line
89.
5
6
Imaginary
axis
90.
2
3
Imaginary
axis
Let z x iy such that:
tan 5
6 y x ⇒
5
θ =
π
6
Real
axis
Let z x iy such that:
tan 2
3 y x ⇒
2
θ =
π
3
Real
axis
y
x 3 3
⇒ y 3
3 x
3 y x ⇒ y 3x
Line
Line
91.
1 i 3 2 cos
2 3
22 2
2 2
2 i
2 2i
4 i sin 4 3 92.
3 3
cos i sin
4 4
2 2i 6 22 cos
22 6
512 cos 3 3
i sin
2 2
512i
4 i sin
4 6
6 6
cos i sin
4 4
© Houghton Mifflin Company. All rights reserved.
93.
95.
1 i 10 2 cos 3 3
i sin
4 4 10 94.
2 10
30 30
cos i sin
4 4
32 cos
3
2 6 i sin 3
2 6
32 cos 3 3
i sin
2 2
320 i1 32i
23 i 5 2 2 cos
2 2 5
5 5
cos i sin
6 6
64 3
2 1 2 i
323 32i
6 i sin 6 5 96.
1 i 8 2 cos 7 7
i sin
4 4 8
2 8 cos 14 i sin 14
16
41 3i 3 4 2 cos 5 5
i sin
3 3
3
42 3 cos 5 i sin 5
321
32

516 Chapter 6 Additional Topics in Trigonometry
97. 5cos 20 i sin 20 125
2 1253
3 5 3 cos 60 i sin 60
i
2
98.
99.
3cos 150 i sin 150 4 3 4 cos 600 i sin 600
81cos 240 i sin 240
81cos 60 i sin 60
81 2 813 i
2
5 5
cos i sin
4 4 10 cos 25 25
i sin
2 2
cos 12
2 i sin 12
2 cos 2 i sin 2 i
100. 2 cos
2 i sin 2 12 2 12 cos 6 i sin 6 2 12 4096
101.
2cos 1.25 i sin 1.25 4 2 4 cos 5 i sin 5 102.
4.5386 15.3428i
4cos 2.8 i sin 2.8 5 4 5 cos 14 i sin 14
140.02 1014.38i
103.
2cos i sin 8 2 8 cos 8 i sin 8
104.
cos 0 i sin 0 20 cos 0 i sin 0
2561 256
1
105. 3 2i 5 597 122i 106.
5 4i 4 21cos5.2221 i sin5.2221 4
441cos20.8884 i sin20.8884
199 393.55i
107.
109.
111.
4cos 10 i sin 10 6 4 6 cos 60 i sin 60 108.
4096
1
2 3
2 i
3 cos
8 i sin
2048 20483i
8 2 3 2 cos
4 i sin
9
2
2 i2 2
9 2 2 9 2 2i
1 2 1 3i 6 cos 4 4
i sin
3 3 6
cos 8 i sin 8
1
4
110.
3cos 15 i sin 15 4 81cos 60 i sin 60
2 cos
81
2 813 i
2
10 i sin 10 5 32 cos
2 i sin
32i
2
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Section 6.5 Trigonometric Form of a Complex Number 517
112. 2 14 1 i is a fourth root of 2 if
2 2 14 1 i 4 .
2 14 1 i 4 2 14 4 1 i 4
2 1 1 i 4
1 21 i 2 1 i 2
1 22i2i
1 24i 2
1 24 2
113. (a) In trigonometric form we have:
2cos 30 i sin 30
2cos 150 i sin 150
2cos 270 i sin 270
(b) There are three roots evenly spaced around a
circle of radius 2. Therefore, they represent
the cube roots of some number of modulus 8.
Cubing them shows that they are all cube
roots of 8i.
(c) 2cos 30 i sin 30 3 8i
2cos 150 i sin 150 3 8i
2cos 270 i sin 270 3 8i
114. (a) In trigonometric form we have:
3cos 45 i sin 45
3cos 135 i sin 135
3cos 225 i sin 225
3cos 315 i sin 315
(b) There are four roots evenly spaced around a
circle of radius 3. Therefore, they represent the
fourth roots of some number of modulus 81.
Raising them to the fourth power shows that
they are all fourth roots of 81.
(c) 3cos 45 i sin 45 4 81
3cos 135 i sin 135 4 81
3cos 225 i sin 225 4 81
3cos 315 i sin 315 4 81
115. (a) In trigonometric form we have:
cos 120 i sin 120
cos 240 i sin 240
cos 0 i sin 0
(b) These are the three cube roots of 1.
(c) cos 120 i sin 120 3 1
cos 240 i sin 240 3 1
cos 0 i sin 0 3 1
© Houghton Mifflin Company. All rights reserved.
116. (a) In trigonometric form we have:
2cos 30 i sin 30
2cos 90 i sin 90
2cos 150 i sin 150
2cos 210 i sin 210
2cos 270 i sin 270
2cos 330 i sin 330
(b) There are six roots evenly spaced around a
circle of radius 2. Raising them to the sixth
power shows that they are the six sixth roots
of 64.
(c)
2cos 30 i sin 30 6 64
2cos 90 i sin 90 6 64
2cos 150 i sin 150 6 64
2cos 210 i sin 210 6 64
2cos 270 i sin 270 6 64
2cos 330 i sin 330 6 64

518 Chapter 6 Additional Topics in Trigonometry
2 cos
4 i sin 4 1 i
2 cos 5 5
i sin
4 4 1 i
117. 2i 2 cos
2 i sin 2 118.
Square roots:
5i 5 cos
2 i sin
Square roots:
5 cos
2
4 i sin 4 10
2
10
2 i
5 cos 5 5
i sin
4 4 10 10
2 2 i
119.
121.
3i 3 cos 3 3
i sin
2 2 120.
Square roots:
3 cos 3 3
i sin
4 4 6 2 6
2 i
3 cos 7 7
i sin
4 4 6
2 6
2 i
2 2i 22 cos 7 7
i sin
4 4 122.
Square roots:
7 7
8 14 cos i sin
8 8 1.554 0.644i
15 15
8 14 cos i sin
8 8 1.554 0.644i
6i 6 cos 3 3
i sin
2 2
Square roots:
6 cos 3 3
i sin
4 4 3 3i
6 cos 7 7
i sin
4 4 3 3i
2 2i 22 cos
Square roots:
8 14 cos
8 i sin
4 i sin
8
1.554 0.644i
9 9
8 14 cos i sin
8 8 1.554 0.644i
4
2 cos
6 i sin
6 6
2 2
2 i
2 cos 7 7
i sin
6 6 6 2 2
2 i
123. 1 3i 2 cos
3 i sin
124.
3
Square roots:
Square roots:
125. (a) Square roots of 5cos 120 i sin 120:
(b)
5cos 60 i sin 60
5cos 240 i sin 240
(c) 5
2 15 i, 5
2 2 15
2 i
1 3i 2 cos 5 5
i sin
3 3
2 cos 5 5
i sin
6 6 2 3 2 1 2 i
6
2 2
2 i
2 cos 11 11
i sin
6 6 6
2 2
2 i
Imaginary
axis
120 360k
5 cos 2 i sin 120 360k
2 , k 0, 1 3
1
Real
−3 −1 1 3
−3
axis
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Section 6.5 Trigonometric Form of a Complex Number 519
126. (a) Square roots of 16cos 60 i sin 60:
(b)
Imaginary
axis
60 360 k
16 cos 2 i sin 60 360 k
2 , k 0, 1
4cos 30 i sin 30
4cos 210 i sin 210
−6
6
2
−2
2
6
Real
axis
(c)
3
4 2 1 2 i 23 2i
−6
4 3
2 1 2 i 23 2i
127. (a) Fourth roots of 16 cos 4 4
i sin (b)
3 3 :
43 2k
416 cos 4 i sin 43 2k
4 , k 0, 1, 2, 3 1
−3 −1
3
−3
2 cos
3 i sin
2 cos 5 5
i sin
6 6
2 cos 4 4
i sin
3 3
2 cos 11 11
i sin
6 6
(c) 1 3i, 3 i, 1 3i, 3 i
Imaginary
axis
3
3
Real
axis
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128. (a) Fifth roots of
k 0, 1, 2, 3, 4
k 0: 2 cos
6 i sin
32 cos 5 5
i sin
6 6 : (b)
532 cos
56 2k
5 i sin 56 2k
5
6
k 1: 2 cos 17 17
i sin
30 30
k 2: 2 cos 29 29
i sin
30 30
k 3: 2 cos 41 41
i sin
30 30
k 4: 2 cos 53 53
i sin
30 30
−3
Imaginary
axis
3
−3
3
Real
axis
(c)
1.732 i, 0.4158 1.956i, 1.989 0.2091i,
0.8135 1.827i, 1.486 1.338i

520 Chapter 6 Additional Topics in Trigonometry
129. (a) Cube roots of 27i 27 cos 3 3
i sin (b)
2 2 :
27 13 cos 32 2k
3 i sin 32 2k
3 , k 0, 1, 2 2
1
Real
3 cos
2 i sin
3 cos 7 7
i sin
6 6
3 cos 11 11
i sin
6 6
(c) 3i, 33
2
2
3 33
i, 3 2 2 2 i
−4
Imaginary
axis
4
−2
−4
2 4
axis
130. (a) Fourth roots of 625i 625 cos
(b)
2 i sin
4625 cos
2 2k
4 i sin 2 2k
4
k 0, 1, 2, 3
k 0: 5 cos
8 i sin
8
k 1: 5 cos 5 5
i sin
8 8
2 :
−6
Imaginary
axis
6
−2
−4
−6
2 4 6
Real
axis
k 2: 5 cos 9 9
i sin
8 8
k 3: 5 cos 13 13
i sin
8 8
(c) 4.619 1.913i, 1.913 4.619i, 4.619 1.913i, 1.913 4.619i
131. (a) Cube roots of 125
(b)
2 1 3i 125 4 4
cos i sin
3 3 :
3125 cos
43 2k
3 i sin 43 2k
5 cos 4 4
i sin
9 9
5 cos 10 10
i sin
9 9
5 cos 16 16
i sin
9 9
(c) 0.8682 4.9240i, 4.6985 1.7101i, 3.8302 3.2139i
3 , k 0, 1, 2 2
−6
−2
−4
−6
Imaginary
axis
6
2 4 6
Real
axis
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Section 6.5 Trigonometric Form of a Complex Number 521
132. (a) Cube roots of
38 cos
34 2k
3 i sin 34 2k
3
k 0, 1, 2
k 0: 2 cos
4 i sin
421 i 8 cos 3 3
i sin
4
4
k 1: 2 cos 11 11
i sin
12 12
k 2: 2 cos 19 19
i sin
12 12
(c) 1.414 1.414i, 1.932 0.5176i, 0.5176 1.9319i
4 : (b) 3
−3
−3
Imaginary
axis
3
Real
axis
133. (a) Cube roots of 64i 64 cos
(b)
2 i sin
64 13 cos 2 2k
3 i sin 2 2k
4 cos
6 i sin
6
4 cos 5 5
i sin
6 6
4 cos 9 9
i sin
6 6 4
(c) 23 2i, 23 2i, 4i
3
cos
2
2 :
5
1
−3 −2 −1 1 2 3
3 , k 0, 1, 2
i sin
3
2
Imaginary
axis
5
3
2
−2
−3
−5
Real
axis
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134. (a) Fourth roots of i cos
2 i sin 2 : (b) Imaginary
axis
2 2k
41 cos 4 i sin 2 2
2k
4
Real
k 0, 1, 2, 3
k 0: cos
8 i sin 8
k 1: cos 5 5
i sin
8 8
k 2: cos 9 9
i sin
8 8
k 3: cos 13 13
i sin
8 8
(c) 0.9239 0.3827i, 0.3827 0.9239i, 0.9239 0.3827i, 0.3827 0.9239i
−2 2
−2
axis

522 Chapter 6 Additional Topics in Trigonometry
135. (a) Fifth roots of 1 cos 0 i sin 0:
(b)
cos 2k 2k
i sin
5 5 , k 0, 1, 2, 3, 4 2
Real
cos 0 i sin 0
cos 2 2
i sin
5 5
cos 4 4
i sin
5 5
cos 6 6
i sin
5 5
cos 8 8
i sin
5 5
(c) 1, 0.3090 0.9511i, 0.8090 0.5878i, 0.8090 0.5878i, 0.3090 0.9511i
Imaginary
axis
−2 2
−2
axis
136. (a) Cube roots of 1000 1000cos 0 i sin 0: (b)
31000 cos 2k 2k
i sin
3 3
k 0, 1, 2
k 0: 10cos 0 i sin 0 10
k 1: 10 cos 2 2
i sin
3 3
k 2: 10 cos 4 4
i sin
3 3
(c) 10, 5 53i, 5 53i
8
6
4
−8 −6 −4 −2
Imaginary
axis
−6
−8
2 4 6 8
Real
axis
137. (a) Cube roots of 125 125cos 180 i sin 180 are: (b)
3125 cos
180 360k
3 i sin 180 360k
5cos 60 i sin 60
5cos 180 i sin 180
5cos 300 i sin 300
(c) 5 2 53
2 i, 5, 5 2 53
2 i
138. (a) Fourth roots of 4 4cos 180 i sin 180:
2cos 45 i sin 45
2cos 135 i sin 135
2cos 225 i sin 225
2cos 315 i sin 315
(c) 1 i, 1 i, 1 i, 1 i
3 , k 0, 1, 2 −4 −3 −2 −1
−2
(b)
Imaginary
axis
4
3
2
1
−3
−4
Imaginary
axis
2
2 3 4
−2 2
−2
Real
axis
Real
axis
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Section 6.5 Trigonometric Form of a Complex Number 523
139. (a) Fifth roots of 1281 i 1282cos 135 i sin 135 are: (b)
(c)
22cos 27 i sin 27 22 cos 3 3
i sin
20 20
22cos 99 i sin 99
22cos 171 i sin 171
22cos 243 i sin 243
22cos 315 i sin 315
2.52 1.28i, 0.44 2.79i, 2.79 0.44i, 1.28 2.52i, 2 2i
Imaginary
axis
−2 −1 1 2
−2
Real
axis
140. (a) Sixth roots of 729i 729 cos
2 i sin
6729
2 k
cos 2
6 i sin 2 2 k
6 , k 0, 1, 2, 3, 4, 5
3 cos
12 i sin 12
3 cos 5 5
i sin
12 12
2 :
(b)
−4
Imaginary
axis
−1
5
4
2
−4
−5
2
4
5
Real
axis
3 cos 9 9
i sin
12 12
3 cos 13 13
i sin
12 12
3 cos 17 17
i sin
12 12
3 cos 21 21
i sin
12 12
(c)
2.898 0.776i, 0.776 2.898i, 2.121 2.121i,
2.898 0.776i, 0.776 2.898i, 2.121 2.121i
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141.
x 4 i 0
x 4 i
The solutions are the fourth roots of i cos
:
2 i sin 2
2 2k
41 cos 4 i sin 2 2k
4 , k 0, 1, 2, 3
cos
8 i sin 8
cos 5 5
i sin
8 8
cos 9 9
i sin
8 8
Imaginary
axis
2
−2 2
Real
axis
cos 13 13
i sin
8 8
−2

524 Chapter 6 Additional Topics in Trigonometry
142.
x 3 27 0
x 3 27
Solutions are cube roots of 27 27cos i sin :
2k
3cos i sin 3
2k
3 cos i sin
3
3 , k 0, 1, 2 2
3 cos
3 i sin 3 3 2 33
1
2 i
Real
−4 −2 −1 1 2 4 axis
3 cos 5 5
i sin
3 3 3 2 33
2 i
Imaginary
axis
4
−2
−4
143.
x 5 243
The solutions are the fifth roots of 243 243cos i sin :
5243 cos 2k
5 i sin
2k
5
, k 0, 1, 2, 3, 4
3 cos
5 i sin
5
3 cos 3 3
i sin
5 5
3 cos 5 5
i sin
5 5 3
−5−4
−2 −1
Imaginary
axis
5
4
1
2
4
5
Real
axis
3 cos 7 7
i sin
5 5
−4
−5
3 cos 9 9
i sin
5 5
144.
x 4 81 81cos 0 i sin 0
3 cos 2k 2k
i sin
4 4 , k 0, 1, 2, 3 5
4
2
1
3cos 0 i sin 0 3
−4 −1 1 2 4 5
3 cos
2 i sin 2 3i
−2
3cos i sin 3
3 cos 3 3
i sin
2 2 3i
Imaginary
axis
−4
−5
Real
axis
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Section 6.5 Trigonometric Form of a Complex Number 525
145.
x 4 16i
The solutions are the fourth roots of
16i 16 cos 3 3
i sin
2 2 :
416 cos
32 2k
4 i sin 32 2k
4 , k 0, 1, 2, 3
2 cos
3
8 i sin 3
8
Imaginary
axis
2 cos
7
8 i sin 7
8
2 cos
11
8 i sin 11
8
−3
−1
3
1
3
Real
axis
2 cos
15
8 i sin 15
8
3
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146.
147.
x 6 64i 0
x 6 64i
The solutions are the sixth roots of 64i:
2 2k
664 cos 6 i sin 2 2k
6 , k 0, 1, 2, 3, 4, 5
k 0: 2 cos
12 i sin 1.932 0.5176i
12
k 1: 2 cos 5 5
i sin 0.5176 1.932i
12 12
Real
k 2: 2 cos 3 3
i sin
4 4 1.414 1.414i
k 3: 2 cos 13 13
i sin
12 12 1.932 0.5176i
k 4: 2 cos 17 17
i sin
12 12 0.5176 1.932i
k 5: 2 cos 7 7
i sin
4 4 1.414 1.414i
x 3 1 i 0
x 3 1 i 2cos 315 i sin 315
The solutions are the cube roots of 1 i:
315 360k
32 cos 3 i sin 315 360k
3 , k 0, 1, 2
62cos 105 i sin 105
62cos 225 i sin 225
62cos 345 i sin 345
Imaginary
axis
3
−3 3
−3
Imaginary
axis
2
−2 2
−2
axis
Real
axis

526 Chapter 6 Additional Topics in Trigonometry
148.
x 4 1 i 0
x 4 1 i 2cos 225 i sin 225
Imaginary
axis
2
The solutions are the fourth roots of 1 i:
225 360k
42 cos 4 i sin 225 360k
4
−2 2
Real
axis
k 0, 1, 2, 3
−2
k 0: 8 2cos 56.25 i sin 56.25 0.6059 0.9067i
k 1: 8 2cos 146.25 i sin 146.25 0.9067 0.6059i
k 2: 8 2cos 236.25 i sin 236.25 0.6059 0.9067i
k 3: 8 2cos 326.25 i sin 326.25 0.9067 0.6059i
149.
E I Z 150.
10 2i4 3i
40 6 30 8i
34 38i
E I Z
12 2i3 5i
36 10 60 6i
26 66i
151.
Z E I
152.
Z E I
5 5i
2 4i 2 4i
2 4i
4 5i 10 2i
10 2i 10 2i
10 20 10 20i
4 16
40 10 50 8i
100 4
30 10i
20
50 42i
104
3 2 1 2 i
25
52 21
52 i
153.
I E Z
12 24i 12 20i
12 20i 12 20i
144 480 288 240i
144 400
624 48i
544
39
34 3 34 i
154.
I E Z
15 12i 25 24i
25 24i 25 24i
375 288 300 360i
625 576
663 60i
1201
663
1201 60
1201 i
155. True.
1
2 1 3i 9
1
2 3
2 i 9 1 156. False. 3 i 2 2 23i 8i
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Section 6.5 Trigonometric Form of a Complex Number 527
157. True
158.
z 1
r 1cos 1 i sin 1
z 2 r 2 cos 2 i sin 2 cos 2 i sin 2
cos 2 i sin 2
r 1
r 2 cos 2 2 sin2 2 cos 1 cos 2 sin 1 sin 2 isin 1 cos 2 sin 2 cos 1
r 1
cos1
r 2 i sin1 2
2
159.
z rcos i sin
160. (a)
zz rcos i sin rcos i sin
z rcos i sin
r 2 cos i sin
rcos i sin
r 2 cos 0 i sin 0
r 2
(b)
z
z rcos i sin
rcos i sin
r r cos i sin
cos 2 i sin 2
161.
z rcos i sin
162. Let a 0 and b in Euler’s Formula:
z rcos i sin
e abi e a cos b i sin b
rcos i sin
e 0i e 0 cos i sin
rcos i sin
ei
1
ei
1 0
163.
d 16 cos
4 t
164. Maximum displacement:
1
16
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165.
Maximum displacement: 16
Lowest possible t- value:
4 t 2 ⇒ t 2
d 1 8 cos12t
Maximum displacement:
Lowest possible t-value: 12t
2 ⇒ t 1
24
1
8
Lowest positive value: t 4 5
166. Maximum displacement:
1
12
Lowest positive value: t 1
60

528 Chapter 6 Additional Topics in Trigonometry
167.
2 cosx 2 cosx 0
4 cos x cos 0
cos x 0
x
2 , 3
2
168.
sin x 3
2 sin x 3
2 0
cos x cos x 0
2 cos x 0
x
2 , 3
2
169.
sin x
1
2 sin x
3 sin x
3 3 2
3 sin x 3 3 2 1 2
cos x sin
3 3 4
cos x
3
2 3 4
cos x 3
2
x 5
6 , 7
6
170.
tanx cos x 5
2 0
tan x sin x 0
sin x
1
cos x 1 0
x 0,
Review Exercises for Chapter 6
1. Given:
b a sin B
sin A
c a sin C
sin A
A 32, B 50, a 16
C 180 32 50 98
16 sin 50
sin 32
16 sin 98
sin 32
23.13
29.90
2. Given:
b a sin B
sin A
c a sin C
sin A
A 38, B 58, a 12
C 180 38 58 84
12 sin 58
sin 38
12 sin 84
sin 38
16.53
19.38
3. Given:
A 180 25 105 50
b c sin B
sin C
a c sin A
sin C
5. Given:
a b sin A
sin B
c b sin C
sin B
B 25, C 105, c 25
25 sin 25
sin 105
25 sin 50
sin 105
10.94
19.83
A 60 15 60.25, B 45 30 45.5, b 4.8
C 180 60.25 45.5 74.25 74 15
4.8 sin 60.25
sin 45.5
4.8 sin 74.25
sin 45.5
5.84
6.48
4. Given:
A 180 20 115 45
b c sin B
sin C
a c sin A
sin C
B 20, C 115, c 30
30 sin 20
sin 115
30 sin 45
sin 115
11.32
23.41
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Review Exercises for Chapter 6 529
6. Given:
C 180 82.75 28.75 68.5 68 30
a b sin A
sin B
c b sin C
sin B
7. Given:
sin B b sin A
a
No solution
A 82 45 82.75, B 28 45 28.75, b 40.2
40.2 sin 82.75
sin 28.75
40.2 sin 68.5
sin 28.75
16.5 sin 75
2.5
82.91
77.76
A 75, a 2.5, b 16.5
6.375 ⇒ no triangle formed
8. Given:
sin B b sin A
a
c a sin C
sin A
A 15, a 5, b 10
10 sin 15
5
Case 1: B 31.2
Case 2: B 148.8
C 180 15 31.2 133.8
5 sin 133.8
sin 15
0.5176 ⇒ B 31.2 or 148.8
13.94
C 180 15 148.8 16.2
c a sin C
sin A
5 sin 16.2
sin 15
5.4
9. Given:
sin A a sin B
b
C 180 115 34.2 30.8
c
B 115, a 9, b 14.5
9 sin 115
14.5
0.5625 ⇒ A 34.2
b
14.5
sin C sin 30.8 8.18
sin B sin 115
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10. Given:
sin A a sin B
b
No solution
11. Given:
sin A a sin C
c
Case 1:
A 60.5
B 150, a 64, b 10
64 sin 150
10
C 50, a 25, c 22
25 sin 50
22
B 180 50 60.5 69.5
b c sin B
sin C 220.9367 26.90
0.7660
3.2 ⇒ no triangle formed
250.7660
22
0.8705 ⇒ A 60.5 or 119.5
Case 2:
A 119.5
B 180 50 119.5 10.5
b c sin B
sin C 5.24

530 Chapter 6 Additional Topics in Trigonometry
12. Given:
sin A a sin B
b
C 180 25 40.9 114.1
c b sin C
sin B
B 25, a 6.2, b 4
6.2 sin 25
4
Case 1: A 40.9
Case 2: A 139.1
4 sin 114.1
sin 25
0.6551 ⇒ A 40.9 or 139.1
8.64
C 180 25 139.1 15.9
c b sin C
sin B
4 sin 15.9
sin 25
2.60
13.
A 27, b 5, c 8 14.
B 80, a 4, c 8 15.
C 122, b 18, a 29
Area 1 2bc sin A
Area 1 2ac sin B
Area 1 2ab sin C
1 258sin 27
1 2480.9848
1 22918 sin 122
9.08 square units
15.76 square units
221.34 square units
16.
Area 1 2ab sin C
17.
h 50 tan 17 15.3 meters
1 212074 sin 100
4372.5 square units
17°
50
h
18. In triangle ABC, A 90 62 28, B 90 38 128, and C 180 A B 24.
b c sin B
sin C
5 sin 128
sin 24
9.6870
h b sin A b sin 28 4.548 4.5 miles
C
h
b
B
5
A
19.
sin 28 h 75
20.
a
sin 75 400
sin 37.5
h 75 sin 28 35.21 feet
cos 28 x
75
tan 45 H x
H x tan 45 66.22 feet
Height of tree:
75
45°
x
28°
x 75 cos 28 66.22 feet
H h 31 feet
h
H
w
22° 30′
a
sin 67.5 w a
w 634.7 sin 67.5 586.4 feet
Tree
37° 30′
a
400 sin 75
sin 37.5
67° 30′ 75°
400
15°
634.7 feet
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Review Exercises for Chapter 6 531
21. Given:
a 18, b 12, c 15
cos A b2 c 2 a 2
2bc
cos B a2 c 2 b 2
2ac
C 180 A B 55.77
122 15 2 18 2
21215
182 15 2 12 2
21815
0.125 ⇒ A 82.82
0.75 ⇒ B 41.41
22. Given:
a 10, b 12, c 16
cos C a2 b 2 c 2
2ab
cos B a2 c 2 b 2
2ac
A 180 B C 38.62
102 12 2 16 2
21012
102 16 2 12 2
21016
0.05 ⇒ C 92.87
0.6625 ⇒ B 48.51
23. Given:
sin A a sin C
c
a 9, b 12, c 20 24.
cos C a2 b 2 c 2
2ab
81 144 400
2912
0.8102 ⇒ C 144.1
9 sin144.1
20
0.264 ⇒ A 15.3
B 180 144.1 15.3 20.6
a 7, b 15, c 19
cos C a2 b 2 c 2
2ab
cos B a2 c 2 b 2
2ac
A 180 B C 19.6
0.4143 ⇒ C 114.5
0.6955 ⇒ B 45.9
© Houghton Mifflin Company. All rights reserved.
25. Given:
cos A b2 c 2 a 2
2bc
cos B a2 c 2 b 2
2ac
C 180 A B 145.83
26. Given:
a 6.5, b 10.2, c 16
a 6.2, b 6.4, c 2.1
cos A b2 c 2 a 2
2bc
cos B a2 c 2 b 2
2ac
C 180 A B 19.10
10.22 16 2 6.5 2
210.216
6.52 16 2 10.2 2
26.516
6.42 2.1 2 6.2 2
26.42.1
6.22 2.1 2 6.4 2
26.22.1
0.97 ⇒ A 13.19
0.93 ⇒ B 20.98
0.26 ⇒ A 75.06
0.07 ⇒ B 85.84

532 Chapter 6 Additional Topics in Trigonometry
27. Given:
c 2 a 2 b 2 2ab cos C 25 2 12 2 22512 cos 65 515.4290 ⇒ c 22.70
sin A a sin C
c
C 65, a 25, b 12
25 sin 65
22.70
B 180 A C 28.62
0.998 ⇒ A 86.38
28. Given:
sin C c sin B
b
B 48, a 18, c 12
b 2 a 2 c 2 2ac cos B 18 2 12 2 21812 cos 48 178.9356 ⇒ b 13.38
A 180 B C 90.19
(Answers may vary.)
12 sin 48
13.38
0.666663 ⇒ C 41.81
29. Given:
b 2 a 2 c 2 2ac cos B 16 16 244cos 110 42.94 ⇒ b 6.55
sin A a sin B
b
B 110, a 4, c 4
4 sin 110
6.55
c a ⇒ C A 35
0.5739 ⇒ A 35
30. Given:
b 2 a 2 c 2 2ac cos B 100 400 4000.8660 846.4 ⇒ b 29.09
sin C c sin B
b
sin A a sin B
b
B 150, a 10, c 20
200.5
29.09
100.5
29.09
0.3437 ⇒ C 20.1
0.1719 ⇒ A 9.9
31. Given:
b 2 a 2 c 2 2ac cos B 12.4 2 18.5 2 212.418.5 cos 55.5 236.1428 ⇒ b 15.37
sin A a sin B
b
B 55 30 55.5, a 12.4, c 18.5
12.4 sin 55.5
15.37
C 180 A B 82.82 82 49
32. Given:
33.
sin A a sin B
b
24.2 sin 85.25
35.61
C 180 A B 52.12 52 7
0.665 ⇒ A 41.68 41 41
B 85 15 85.25, a 24.2, c 28.2
b 2 a 2 c 2 2ac cos B 24.2 2 28.2 2 224.228.2 cos 85.25 1267.8567 ⇒ b 35.61
0.677 ⇒ A 42.63 42 38
a 2 5 2 8 2 258 cos 152 159.6 ⇒ a 12.63 ft
b 2 5 2 8 2 258 cos 28 18.36 ⇒ b 4.285 ft
a
5
8
28°
8
5 b
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Review Exercises for Chapter 6 533
34.
a 2 15 2 20 2 21520 cos146 1122.42 ⇒ a 33.5 m
b 2 15 2 20 2 21520 cos34 127.58 ⇒ b 11.3 m
b
15
20
a
a
20
34°
15
b
35. Angle between planes is 5 67 72.
In two hours, distances from airport are 850 miles and 1060 miles.
By the Law of Cosines,
d 2 850 2 1060 2 28501060 cos72 1,289,251.376
72°
d 1135.5 miles.
850
d
1060
36.
b 2 a 2 c 2 2ac cos B
37.
a 4, b 5, c 7
300 2 425 2 2300425 cos180 65
378,392.66
b 615.1 meters
s a b c
2
Area ss as bs c
8431
4 5 7
2
8
9.798 square units
38.
a 15, b 8, c 10
39.
a 64.8, b 49.2, c 24.1
s
15 8 10
2
16.5
s a b c
2
64.8 49.2 24.1
2
69.05
Area 16.51.58.56.5
Area ss as bs c
36.98 square units
69.054.2519.8544.95
511.7 square units
© Houghton Mifflin Company. All rights reserved.
40.
s
8.55 5.14 12.73
2
Area ss as bs c
13.214.668.070.48
15.4 square units
13.21
41. Initial point:
Terminal point:
5, 4
2, 1
v 2 5, 1 4 7, 5
42. Initial point: 0, 1 43. Initial point: 0, 10
Terminal point: 6, 7 2
Terminal point: 7, 3
v 6 0, 7 2 1 6, 5 2
v 7 0, 3 10 7, 7
44. Initial point: 1, 5 45. 8 cos 120i 8 sin 120j 4, 43
Terminal point: 15, 9
v 15 1, 9 5 14, 4
46. 1 2 cos 225, 1 2 sin 225 2 4 , 2 4

534 Chapter 6 Additional Topics in Trigonometry
47.
2u 48.
2u v
y
y
y
2u + v
2u
v
2u
u
x
1 2 v 49. x
1
− v
2
x
v
50.
u 2v 51.
u 2v 52.
v 2u
y
y
y
u + 2v
u
v
x
u
2v
x
u − 2v
−2v
x
−2u
v − 2u
53. (a) u v 1, 3 3, 6 4, 3
(b) u v 2, 9
(c) 3u 3, 9
(d) 2v 5u 6, 12 5, 15 11, 3
54. (a) u v 4, 5 0, 1 4, 4
(b) u v 4, 6
(c) 3u 12, 15
(d) 2v 5u 0, 2 20, 25 20, 23
55. (a) u v 5, 2 4, 4 1, 6
(b) u v 9, 2
(c) 3u 15, 6
(d) 2v 5u 8, 8 25, 10 17, 18
56. (a) u v 1, 8 3, 2 4, 10
(b) u v 2, 6
(c) 3u 3, 24
(d) 2v 5u 6, 4 5, 40 11, 44
57. (a) u v 2, 1 5, 3 7, 2
(b) u v 3, 4
(c) 3u 6, 3
(d) 2v 5u 10, 6 10, 5 20, 1
59. (a) u v 4, 0 1, 6 3, 6
(b) u v 5, 6
(c) 3u 12, 0
(d) 2v 5u 2, 12 20, 0 18, 12
58. (a) u v 0, 6 1, 1 1, 5
(b) u v 1, 7
(c) 3u 0, 18
(d) 2v 5u 2, 2 0, 30 2, 28
60. (a) u v 7, 3 4, 1 3, 4
(b) u v 11, 2
(c) 3u 21, 9
(d) 2v 5u 8, 2 35, 15 27, 17
© Houghton Mifflin Company. All rights reserved.

Review Exercises for Chapter 6 535
61.
3v 310i 3j 30i 9j 30, 9 62.
v 10i 3j
y
1
2 v 5i 3 2 j
30
y
25
20
15
10
5
5 10 15 20 25 30
x
5
4
3
2
1
−5 −4 −3 −2 −1
−2
−3
−4
−5
1
2
3
4
5
x
63.
w 4u 5v 46i 5j 510i 3j 64.
50
40
30
20
10
−10
−20
−30
−40
−50
y
74i 5j
10 20 30 40 90
x
3v 2u 310i 3j 26i 5j 18i 19j
y
21
18
15
12
9
6
3
x
3 6 9 12 15 18 21
65.
u 6
Unit vector: 1 60, 6 0, 1
66.
v 12 2 5 2 169 13
Unit vector: 12
13 , 13
5
67.
v 5 2 2 2 29
1
Unit vector: 5, 2 5
29 , 2
29 29
68.
w 7
Unit vector: 1 7i i
7
69. u 1 8, 5 3 9, 8 9i 8j
70. u 6.4 2, 10.8 3.2 8.4i 14j
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71.
72.
v 10i 10j
v 10 2 10 2 200 102
tan 10
since v is in Quadrant II.
10 1 ⇒
v 102cos 135i sin 135j
135
v 4i j
v 4 2 1 2 17
tan 1
4 1 4 ⇒
v 17cos 346i sin 346 j
346,
since
is in Quadrant IV.

536 Chapter 6 Additional Topics in Trigonometry
73.
u v 23.1752i 22.9504j
u v 32.62
tan 22.9504
23.1752 ⇒
y
u 15cos 20i sin 20j 74.
v 20cos 63i sin 63j
44.72
y
u 12cos 82i sin 82j
v 8cos12i sin12j
u v 9.4953i 10.2199j
u v 13.95
tan 10.2199
9.4953 ⇒
47.11
30
25
20
15
10
5
44.72°
32.62
−5 5 10 15 20 25 30
−5
x
14
12
10
8
6
4
2 47.11°
13.95
−4 −2 2 10 12
−2
x
75.
F 1 250cos 60, sin 60
F 2 100cos 150, sin 150
F 3 200cos90, sin90
F F 1 F 2 F 3 38.39746, 66.50635
tan 66.50635
38.39746 ⇒
60
F 38.39746 2 66.50635 2 76.8 pounds
76. Force One: u 85i
Force Two: v 50 cos 15i 50 sin 15j
Resultant Force:
u v 85 50 cos 15i 50 sin 15j
u v 85 50 cos 15 2 50 sin 15 2
tan
77. Rope One:
Rope Two:
Resultant:
85 2 8500 cos 15 50 2
133.92 lb
50 sin 15
85 50 cos 15 ⇒
5.5
from the 85-pound force.
u ucos 30i sin 30j u
3
2 i 1 2 j
v ucos 30i sin 30j u 3
2 i 1 2 j
u v uj 180j
u 180
Therefore, the tension on each rope is u 180 pounds.
© Houghton Mifflin Company. All rights reserved.

Review Exercises for Chapter 6 537
78. By symmetry, the magnitudes of the tensions are equal.
T Tcos 120i sin 120j
T sin 120 1 100
200 ⇒ T
2 32 200 115.5 lbs
3
⏐⏐T⏐⏐
60°
⏐⏐T⏐⏐
60°
120°
200 lb
79. Airplane velocity: u 430cos 315, sin 315
Wind velocity: w 35cos 60, sin 60
u w 321.5559, 273.7450
u w 422.3 mph
tanu w 273.7450
321.5559
0.8513 ⇒
40.4
The bearing from the north is 90 40.4 130.4.
80. Airspeed:
Wind: w 32i
Groundspeed u w 394i 3623j
u w 394 2 3623 2 740.5 kmhr
tan 3623
394
Bearing:
u 724cos 60i sin 60j
362i 3j
⇒
57.9
N 32.1 E (or 32.1 in airplane navigation)
y
30°
724
32
x
© Houghton Mifflin Company. All rights reserved.
81. u v 0, 2 1, 10 0 20 20
83. u v 6, 1 2, 5 62 15 7
85.
u u 3, 4 3, 4
9 16 25 u 2
87. 4u v 43, 4 2, 1 46 4 40
89.
91.
u 22, 4, v 2, 1
cos u v
u v
u cos 7 7
i sin
4
0.966 ⇒
8
243 ⇒
v cos 5 5
i sin
6 6 j 3 2 , 1 2
82. u v 4, 5 3, 1 12 5 17
84. u v 8i 7j 3i 4j 24 28 52
86. v 3 2 2 1 2 3 5 3
88. u vu 103, 4 30, 40
90. u 3, 1, v 4, 5
cos u v
u v 17
1041 ⇒
4 j 1 2 , 1 92. Angle 45 60 105
2
165 or 11
12
160.5
cos u v 322 122
u v 11
32.9

538 Chapter 6 Additional Topics in Trigonometry
93.
cos u v
u v
0 ⇒
90
y
4
3
2
1
u
−4 −3 −2 −1
−1
1 2 3 4
−2
−3 v
−4
x
94.
180
y
8
6
4
u
2
−8 −6 −4
v −2
2 4 6 8
−4
−6
x
−8
95.
cos u v 70 15
u v 74109
8
6
4
2
y
0.612
⇒
−4 −2 4 6 8 10 12
−2
v
52.2
x
96.
54.1
y
8
6
u 4
2
−10 −8 −6 −4
−2
2 4
v
−4
−6
−8
x
−4
−6
u
−8
97.
u 39, 12, v 26, 8
u v 3926 128
1110 0 ⇒ u and v are not orthogonal.
v 2 3u ⇒ u and v are parallel.
98. u v 8, 4 5, 10 40 40 0 ⇒ orthogonal
99.
u 8, 5, v 2, 4
u v 82 54
4 0 ⇒ u and v are not orthogonal.
u kv ⇒ u and v are not parallel.
Neither
100. 3 4 v 3 420, 68 15, 51 u ⇒ parallel
101. 1, k 1, 2 1 2k 0 ⇒ k 1 2
102. 2, 1 1, k 2 k 0 ⇒ k 2
103. k, 1 2, 2 2k 2 0 ⇒ k 1 104. k, 2 1, 4 k 8 0 ⇒ k 8
© Houghton Mifflin Company. All rights reserved.

Review Exercises for Chapter 6 539
105.
u 4, 3, v 8, 2 106.
proj v u
u v
v 2v 26
68
u proj v u 4, 3 52
17 , 13
17
16 17 , 64
17
u 52
17 , 13 17 16 17 , 64
17
8, 2 134, 1
17
u 5, 6, v 10, 0
u
proj v u
v 50
v2 v 10, 0 5, 0
100
u 5, 0 0, 6
107.
u 2, 7, v 1, 1 108.
proj v u
u v 5
v2 v
2 1, 1 5 2 , 2
5
u proj v u 2, 7 5 2 , 5 2 9 2 , 9 2
u 5 2 , 5 2 9 2 , 9 2
u 3, 5, v 5, 2
proj v u
u v
v
25
2 v 5, 2
29
u 125 29 , 50
29 38
29 , 95
29
109. 48 inches 4 feet
Work 18,0004 72,000 ft lb
110. Force 500 sin 12104 lbs
111.
5i 5
i 1 Imaginary
Imaginary
axis
axis
3
2
1
5
4
3
2
5i
−3 −2
Real
axis
−1 1 2 3
−1 −i
−2
1
−4−3 −2 −1
−2
−3
1 2 3 4 5
−3
−4
−5
Real
axis
© Houghton Mifflin Company. All rights reserved.
113.
115.
7 5i 72 5 2 74 114.
Imaginary
axis
2
−2
−2
−4
−6
−8
−10
z
4 6
7 − 5i
8 10
Real
axis
z 2 2i 116.
4 4 22
7
4
z 22 cos 7 7
i sin
4 4
3 9i
−3 + 9i
z
−4−3 −2 −1
3
4
Imaginary
axis
9
8
7
6
5
4
2
1
9 81 90 310
1 2 3 4 5
z 2 2i
Real
axis
4 4 22
z 22 cos 3 3
i sin
4 4

540 Chapter 6 Additional Topics in Trigonometry
117.
119.
z 3 i 118.
z 3 1 2
7
6
z 2 cos 7 7
i sin
6 6
z 2i 120.
z 2
3
2
z 2 cos 3 3
i sin
2 2
z 3 i
z 3 1 2
5
6
z 2 cos 5 5
i sin
6 6
z 4i
z 4
2
z 4 cos
2 i sin
2
121.
5
2 cos
2 i sin
122. 6 cos
2
3
2 4 cos
6 i sin 2
4 i sin
3
6 6
4 10
3 3
cos i sin
4 4
5 5
cos i sin
6 6 33 3i
123.
20cos 320 i sin 320
5cos 80 i sin 80
4cos 240 i sin 240
124.
3
9 cos230 95 i sin230 95 1 cos 135 i sin 135
3
2
6 2
6 i
125. (a)
126. (a)
2 2i 22 cos 7 7
i sin
4 4
3 3i 32 cos
(b) 22 cos 7 7
i sin
4
(c) 2 2i3 3i 6 6 12
4 i sin
4
4 32 cos
4 4i 42 cos
4 i sin
4 2
(b) 42 cos
4 i sin
5
cos
4
(c) 4 4i1 i 4 4i 4i 4 8i
4
1 i 2 cos 5 5
i sin
4 4
4 i sin 4 12cos 2 i sin 2 12
i sin
5
4 8
3
cos
2
i sin
3
2 8i
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Review Exercises for Chapter 6 541
127. (a)
(b)
128. (a)
i cos 3 3
i sin
2 2
2 2i 22 cos
4 i sin
3
cos
2
i sin
3
2 22 cos
(c) i2 2i 2i 2 2 2i
(b)
4 cos
2 i sin
4i 4 cos
2 i sin
(c) 4i1 i 4i 4 4 4i
4
4 i sin
1 i 2 cos
4 i sin 4
2
4 22
22
2
2 i 2 2
2 2i
2 2 cos 4 i sin 4 42 cos
7 7
cos i sin
4 4
42
2
2 i2 2
4 4i
4 i sin
4
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129. (a)
(b)
130. (a)
3 3i 32 cos 7 7
i sin
4 4
32 cos 7 7
i sin
4 4
2 2i 22 cos
4 i sin
22 cos
4 i sin
4 3 2
cos
3
2 i sin
3 2 i 3 2 i
3
2
(c) 3 3i 2 2i
2 2i 2 2i 6 12i 6 12i
8
8 3 2 i
(b)
(c)
1 i
2 2i 1 i
21 i 1 2
4
1 i 2 cos 5 5
i sin
4 4
2 2i 22 cos 5 5
i sin
4 4
1 i
2 2i 2
22 cos 0 i sin 0 1 2

542 Chapter 6 Additional Topics in Trigonometry
131.
5 cos
12 i sin
12 4 5 4 4 12
625 cos
3 i sin
625
1
2 3
2 i
625
2 6253 i
2
i sin
4
12 132.
3
2
4 4
cos i sin
15 15 5 2 5
4
cos
3
32 1 2 3
2 i
16 163i
i sin
4
3
133.
2 3i 6 13cos 56.3 i sin 56.3 6 134.
13 3 cos 337.9 i sin 337.9
13 3 0.9263 0.3769i
2035 828i
1 i 8 2cos 315 i sin 315 8
16cos 2520 i sin 2520
16cos 0 i sin 0
16
135.
137.
138.
3 i 2 cos 5 5
i sin
6 6 136.
Square roots:
2 cos 5 5
i sin 0.3660 1.3660i
12 12
2 cos 17 17
i sin
12 12 0.3660 1.3660i
2i 2 cos 3 3
i sin
2 2
Square roots:
2 cos 3 3
i sin
4 4 2 2 2 i2 2 1 i
2 cos 7 7
i sin
4 4 1 i
5i 5 cos 3 3
i sin
2 2
Square roots:
5 cos 7 7
i sin
4 4 10 10
2 2 i
3 i 2 cos 11 11
i sin
6 6
Square roots:
5 cos 3 3
i sin
4 4 5 2 2 2
2 i 10 10
2 2 i
2 cos 11 11
i sin
12 12 1.3660 0.3660i
2 cos 23 23
i sin
12 12 1.3660 0.3660i
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Review Exercises for Chapter 6 543
139.
2 2i 22 cos 5 5
i sin
4 4 140.
Square roots:
5 5
2 34 cos i sin
8 8 0.6436 1.5538i
13 13
2 34 cos i sin
8 8 0.6436 1.5538i
2 2i 22 cos 3 3
i sin
4 4
Square roots:
3 3
2 34 cos i sin
8 8 0.6436 1.5538i
11 11
2 34 cos i sin
8 8 0.6436 1.5538i
141. (a) Sixth roots of 729i 729 cos 3 3
i sin (b)
2 2 :
(c)
6729 cos 32 2k
i sin 32 2k
6
3 cos
4 i sin
3 cos 7 7
i sin
12 12
3 cos 11 11
i sin
12 12
3 cos 5 5
i sin
4 4
3 cos 19 19
i sin
12 12
3 cos 23 23
i sin
12 12
32
2
4
32 i, 0.7765 2.898i, 2.898 0.7765i,
2
6 , k 0, 1, 2, 3, 4, 5 1
−4 −2 1 4 5
32
2
Imaginary
axis
32 i, 0.7765 2.898i, 2.898 0.7765i
2
5
4
−2
−4
−5
Real
axis
© Houghton Mifflin Company. All rights reserved.
142. (a) Fourth roots of 256i 256 cos
(b)
2 i sin
4256 cos 2 2k
i sin 2 2k
4
4 , k 0, 1, 2, 3
4 cos
8 i sin
8
4 cos 5 5
i sin
8 8
4 cos 9 9
i sin
8 8
4 cos 13 13
i sin
8 8
(c) 3.696 1.531i, 1.531 3.696i, 3.696 1.531i, 1.531 3.696i
2 :
−3
Imaginary
axis
−1
5
3
1
−2
−3
−5
1 2
3
5
Real
axis

544 Chapter 6 Additional Topics in Trigonometry
143. (a) Cube roots of 8 8cos 0 i sin 0:
(b)
38 cos
2k
3 i sin 2k
3
2cos 0 i sin 0
3
1
−3 −1
3
2 cos 2 2
i sin
3 3
2 cos 4 4
i sin
3 3
(c) 2, 1 3i, 1 3i
Imaginary
axis
3
Real
axis
144. (a) Fifth roots of 1024 1024cos i sin :
(b)
51024 cos 2k
5 i sin
2k
5 , k 0, 1, 2, 3, 4
Imaginary
axis
5
4 cos
5 i sin
5
4 cos 3 3
i sin
5 5
−3 −2 −1
1
−5
2
3
5
Real
axis
4 cos 5 5
i sin
5 5 4
4 cos 7 7
i sin
5 5
4 cos 9 9
i sin
5 5
(c)
3.236 ± 2.351i, 1.236 ± 3.804i, 4
145.
x 4 256 0
x 4 256 256cos i sin
Imaginary
axis
5
4256 4 cos 2k
4 i sin
4 cos
4 i sin 4 42
2
4 cos 3 3
i sin
4 4 42 2
4 cos 5 5
i sin
4 4 42 2
4 cos 7 7
i sin
4 4 42
2
2k
4
42 i 22 22 i
2
42 i 22 22 i
2
42 i 22 22 i
2
42 i 22 22 i
2
, k 0, 1, 2, 3
3
2
−5 −3 −2 2
−2
−3
−5
3 5
Real
axis
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Review Exercises for Chapter 6 545
146.
2 2k
532 cos 5 i sin 2 2k
5 , k 0, 1, 2, 3, 4
2 cos
10 i sin 10
x 5 32i 32 cos
2 i sin
2 cos
2 i sin 2 2i
2
−4 −3
Imaginary
axis
−1
5
4
3
1
−3
−4
−5
1
3 4
5
Real
axis
2 cos 9 9
i sin
10 10
2 cos 13 13
i sin
10 10
2 cos 17 17
i sin
10 10
147.
x 3 8i 0
x 3 8i
Imaginary
axis
3
8i 8 cos 3 3
i sin
2 2
38i 3 8 cos 32 2k
3
2 cos
2 i sin 2 2i
i sin 32 2k
3 , k 0, 1, 2
−3
1
−1
−3
3
Real
axis
2 cos 7 7
i sin
6 6 3 i
2 cos 11 11
i sin
6 6 3 i
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148.
x 4 81 81cos i sin
481 cos
2k
4
3 cos
4 i sin
4
3 cos 3 3
i sin
4 4
3 cos 5 5
i sin
4 4
3 cos 7 7
i sin
4 4
i sin
2k
4 , k 0, 1, 2, 3
−4
−2
Imaginary
axis
5
4
2
−2
−4
−5
2 4
5
Real
axis

546 Chapter 6 Additional Topics in Trigonometry
149. True 150. False. There may be no solution, one solution, or
two solutions.
151. Length and direction characterize vectors in plane. 152. A and C appear equivalent.
153.
z 1 z 2 2cos i sin 2cos i sin
z 1
z 2
4cos i sin cos i sin
4cos 2 sin 2 4
2cos
cos i sin
cos i sin
cos i sin cos i sin
cos 2 sin 2 2 sin cos i
cos i sin 2
z
1
2 2
z 1 2
4
2cos i sin
i sin
154. (a) Three roots are not shown.
(b) The modulus of each is 2, and the arguments are 120, 210 and 300.
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Practice Test for Chapter 6 547
Chapter 6
Practice Test
For Exercises 1 and 2, use the Law of Sines to find the remaining sides and
angles of the triangle.
1. A 40, B 12, b 100 2. C 150, a 5, c 20
3. Find the area of the triangle: a 3, b 6, C 130
4. Determine the number of solutions to the triangle: a 10, b 35, A 22.5
For Exercises 5 and 6, use the Law of Cosines to find the remaining sides and
angles of the triangle.
5. a 49, b 53, c 38 6. C 29, a 100, b 300
7. Use Heron’s Formula to find the area of the triangle: a 4.1, b 6.8, c 5.5.
8. A ship travels 40 miles due east, then adjusts its course 12 southward. After traveling
70 miles in that direction, how far is the ship from its point of departure?
9. w 4u 7v where u 3i j and v i 2j. Find w.
10. Find a unit vector in the direction of v 5i 3j.
11. Find the dot product and the angle between u 6i 5j and v 2i 3j.
12. v is a vector of magnitude 4 making an angle of 30 with the positive x-axis.
Find v in component form.
13. Find the projection of u onto v given u 3, 1 and v 2, 4.
© Houghton Mifflin Company. All rights reserved.
14. Give the trigonometric form of z 5 5i.
15. Give the standard form of z 6cos 225 i sin 225.
16. Multiply 7 cos 23 i sin 234cos 7 i sin 7.
9 cos 5 5
i sin
4 4
17. Divide
. 18. Find 2 2i
3cos i sin
8 .
19. Find the cube roots of 8 cos
3 i sin 3 . 20. Find all the solutions to x4 i 0.